$\overline{X}$ is the largest subgroup of $\overline{S}$ which is conjugated
successively into $\overline{S}$ by the entries of $\overline{w}$. Write
$\overline{w}=(\overline{g}_{1},\cdots,\overline{g}_{n})$, and for each $i$
let $g_{i}\in\Cal{L}^{+}$ be a representative of $\overline{g}_{i}$ such that
$g_{i}$ is $\uparrow$-maximal with respect to $\Cal{M}$. Set
$w=(g_{1},\cdots,g_{n})$, and let $X$ be the preimage of $\overline{X}$ in $S$
via $\rho_{S}$. Then $X\leq S_{w}$ by I.4.13(e). If
$\overline{X}\in\overline{\Delta}$ then $X\in\Delta$ and
$w\in\mathbb{D}(\Cal{L}^{+}\mid_{\Delta})$. As
$\Cal{L}^{+}\mid\Delta=\Cal{L}$, we have $w\in\mathbb{D}(\Cal{L})$ in this
case, and hence
$\overline{w}\in\mathbb{D}((\Cal{L}^{+}/\Cal{M})\mid_{\overline{D}})$. On the
other hand, if
$\overline{w}\in\mathbb{D}((\Cal{L}^{+}/\Cal{M})\mid_{\overline{D}})$ then
$\overline{X}\in\overline{\Delta}$. Thus:

$Im(\phi)=(\Cal{L}^{+}/\Cal{M})\mid_{\overline{D}}.$

In particular, $Im(\phi)$ is a partial group (and a locality) in this way, and
the mapping

$\phi_{0}:\Cal{L}\to Im(\phi)$

given by $\phi$ is then a homomorphism of partial groups. One observes that
the induced mapping

$\phi_{0}^{*}:\mathbb{W}(\Cal{L})\to\mathbb{W}(Im(\phi))$

of free monoids carries $\mathbb{D}(\Cal{L})$ onto $\mathbb{W}(Im(\phi))$.
That is, $\phi_{0}$ is a projection. Then, by I.5.6 there is an isomorphism
$\overline{\phi}:\Cal{L}/\Cal{N}\to Im(\phi)$ induced by $\phi$.

Let $\eta:\Cal{L}/\Cal{N}\to\Cal{L}^{+}/\Cal{M}$ be the composition of
$\overline{\phi}$ followed by the inclusion map of $Im(\phi)$ into
$\Cal{L}^{+}/\Cal{M}$. Our aim is to now show that $\Cal{L}^{+}/\Cal{M}$ is an
expansion of $\Cal{L}/\Cal{N}$ via $\eta$. For this, what is required (cf.
definition 4.1) is: (A) $\eta$ is injective (obvious, since $\eta$ is a
composite of injective mappings); (B) $Im(\eta)$ is the restriction of
$\Cal{L}^{+}/\Cal{M}$ to $\Delta\phi$ (which has been verified above); and (C)
$\Delta\phi$ is closed in the fusion system
$\Cal{F}^{\prime}=\Cal{F}_{\overline{S}}(\Cal{L}^{+}/\Cal{M})$. In order to
verify (C) we note first of all that $\Cal{F}=\Cal{F}_{S}(\Cal{L}^{+})$ by
4.3(a). Since $\rho$ and $\phi$ are projections, (C) then follows from 1.10.
Thus $\eta$ defines an expansion, as claimed. Moreover, $\eta$ defines an
elementary expansion, since 1.10 shows that the set $\\{\overline{P}\mid
P\in\Delta^{+}\\}$ of objects of $\Cal{L}^{+}/\Cal{M}$ is the union of
$\overline{D}$ with the set of
$\Cal{F}_{\overline{S}}(\Cal{L}^{+}/\Cal{M})$-conjugates of $\overline{R}$.

We have also to verify that Hypothesis 4.2 holds with respect to
$\Cal{L}/\Cal{M}$ and $\overline{R}$, in order to apply Theorem 4.3 to the
expansion $\Cal{L}^{+}/\Cal{M}$ of $Im(\phi)$. Set $T=S\cap\Cal{N}$, and
suppose first that $T\nleq R$. Then $TR\in\Delta$, so
$\overline{R}\in\overline{\Delta}$, and then 4.2 holds trivially. On the other
hand, suppose that $T\leq R$. Then every subgroup $\overline{Q}$ of
$\overline{S}$ which properly contains an
$\Cal{F}_{\overline{S}}(\Cal{L}^{+}/\Cal{M})$-conjugate of $\overline{R}$ is
the image under $\phi$ of a subgroup $Q$ of $S$ which properly contains an
$\Cal{F}$-conjugate of $R$. That is, 4.2(1) obtains. Now let $\overline{g}\in
N_{\Cal{L}^{+}/\Cal{M}}(\overline{R})$, and let $g\in\Cal{L}^{+}$ be a
$\rho$-preimage of $g$ such that $g$ is $\uparrow$-maximal with respect to
$\Cal{M}$. Then $g\in N_{\Cal{L}^{+}}(R)$ by I.4.13(e). Here
$N_{\Cal{L}^{+}}(R)=N_{\Cal{L}}(R)$ by 4.3 applied to $\Cal{L}^{+}$, so we
have shown that $\rho$ maps $N_{\Cal{L}}(R)$ onto
$N_{\Cal{L}^{+}/\Cal{M}}(\overline{R})$. This verifies 4.2(2).

The preceding analysis applies with $\Cal{N}^{+}$ in the role of $\Cal{M}$,
where we now observe that $\Cal{N}^{+}\leq\Cal{M}$ since
$\Cal{N}\subseteq\Cal{M}$. Let
$\eta_{0}:\Cal{L}/\Cal{N}\to\Cal{L}^{+}/\Cal{N}^{+}$ be the homomorphism which
defines the expansion of $\Cal{L}/\Cal{N}$ to $\Cal{L}^{+}/\Cal{N}^{+}$. By
Theorem 4.3(b) there is then a unique homomorphism
$\beta:\Cal{L}/\Cal{N}^{+}\to\Cal{L}/\Cal{M}$ such that
$\eta_{0}\circ\beta=\eta$, and $\beta$ is in fact an isomorphism. On the other
hand, let $\gamma:\Cal{L}/\Cal{N}^{+}\to\Cal{L}/\Cal{M}$ be the mapping which
sends a maximal coset $[g]_{\Cal{N}^{+}}$ of $\Cal{N}^{+}$ in $\Cal{L}^{+}$ to
the unique maximal coset $[g]_{\Cal{M}}$ containing it. One easily verifies
that $\gamma$ is a homomorphism and that $\eta_{0}\circ\gamma=\eta$, and thus
$\gamma$ is an isomorphism. For $g\in\Cal{M}$ we then have
$[g]_{\Cal{N}^{+}}=\Cal{M}$, and this shows that $\Cal{M}=\Cal{N}^{+}$. That
is, 5.1(c) holds, and we have obtained point (a) of the lemma. Point (b) was
obtained in the course of proving (a). ∎

The remainder of this section will be devoted to the proof of 5.1(b).

###### Definition (5.4) Notation

Set $M=N_{\Cal{L}}(R)$, and $K=M\cap\Cal{N}$. As in the preceding section, for
each $g\in\Cal{L}^{+}$ set

$\Cal{U}_{g}=\\{U\in R^{\Cal{F}}\mid U\leq S_{g}\\},$

and for each $U\in R^{\Cal{F}}$ set

$\mathbb{Y}_{U}=\\{y\in\Cal{L}\mid R^{y}=U,\ N_{S}(U)^{y^{-1}}\leq
N_{S}(R)\\}.$

We continue to write $\mathbb{X}$ for $\Cal{N}^{\Cal{L}^{+}}$.

The notation (5.4) will remain fixed until the proof of Theorem 5.1 is
complete. Note that $M$ is a subgroup of $\Cal{L}$ by 4.2(2). Then $K$ is a
normal subgroup of $M$ by I.1.8(c).

###### Lemma 5.5

Suppose $T\leq R$. Then $\Cal{N}^{*}=\mathbb{X}$, and
$\mathbb{N}^{*}\cap\Cal{L}=\Cal{N}$.

###### Demonstration Proof

Let $f\in\Cal{N}$, and suppose that $S_{f}$ contains an $\Cal{F}$-conjugate of
$R$. Then $T\leq S_{f}$ since, by I.4.1(a) $T$ is weakly closed in $\Cal{F}$.
Then I.4.1(b) yields the following result.

Let $f^{\prime}\in\mathbb{X}$, and let $f\in\Cal{N}$ and $g\in\Cal{L}^{+}$
such that $f^{\prime}=f^{g}$. That is, assume that
$v:=(g^{-1},f,g)\in\mathbb{D}^{+}$ and that $f^{\prime}=\Pi^{+}(v)$. If
$v\in\mathbb{D}$ then $f^{\prime}=\Pi(v)\in\Cal{N}$. On the other hand,
suppose that $v\notin\mathbb{D}$. Then $S_{v}$ is an $\Cal{F}$-conjugate of
$R$ by 4.2(1). Set $U=(S_{v})^{g^{-1}}$. Then $U=U^{f}$ by (1). Now choose
$a\in\mathbb{Y}_{U}$, and set $v^{\prime}=(g^{-1},a^{-1},a,f,a^{-1},a,g)$.
Then $v^{\prime}\in\mathbb{D}^{+}$ via $U^{g}$, and
$\Pi^{+}(v)=\Pi^{+}(v^{\prime})$. Notice that (1) implies that
$(a,f,a^{-1})\in\mathbb{D}$ via $P:=N_{S_{f}}(U)$, and that

$T@>a>>U@>f>>U@>a^{-1}>>T,$

so that $f^{a^{-1}}\in M$. Thus $f^{a^{-1}}\in K$, and
$\Pi^{+}(v)=\Pi^{+}(g^{-1}a^{-1},f^{a^{-1}},ag)$. This shows:

Assume now that we have $v=(g^{-1},f,g)$ as in (2) (so that $f\in K$), and let
$A$ be an $\Cal{F}$-conjugate of $R$ contained in $S_{v}$. In order to analyze
these things further, we shall need to be able to compute products in
$\Cal{L}^{+}$ in the manner described in 4.14 and 4.15. To that end, note
first of all that since $A^{g}=T$ there exists a unique $h\in M$ and
$y\in\mathbb{Y}_{A}$ such that $g\approx(\mathbb{1},h,y)\in\Phi$, by 4.8. Set
$\phi=(\mathbb{1},h,y)$ and set $\psi=(\mathbb{1},f,\mathbb{1})$. Then
$(\phi^{-1},\psi,\phi)$ is a $\Gamma$-form of $(g^{-1},f,g)$, as defined in
4.13. We then compute via 4.14 that

$f^{\prime}=\Pi^{+}(g^{-1},f,g)=[y^{-1},h^{-1},\mathbb{1}][\mathbb{1},f,\mathbb{1}][\mathbb{1},h,y]=[y^{-1},f^{h},y].$

If $f^{\prime}\in\Cal{L}$ then $(y^{-1},k,y)\in\mathbb{D}$ by 4.11, and so
$f^{\prime}\in\Cal{N}$. Thus:

Now let
$w=(f_{1}^{\prime},\cdots,f_{n}^{\prime})\in\mathbb{W}(\mathbb{X})\cap\mathbb{D}^{+}$,
and set $B=S_{w}$. Suppose that $\Pi^{+}(w)\notin\mathbb{X}$. Then
$\Pi^{+}(w)\notin\Cal{N}$, so $\Pi^{+}(w)\notin\Cal{L}$ by (4). Thus
$w\notin\mathbb{D}$, so $B\in R^{\Cal{F}}$, and then (2) shows that each
$f_{i}^{\prime}$ normalizes $B$. Fix $b\in\mathbb{Y}_{B}$. Then (3) implies
that there exist elements $k_{i}\in K$ such that
$f_{i}^{\prime}=[b^{-1},k_{i},b]$. One observes that the sequence of elements
$(b^{-1},k_{i},b)$ of $\Phi$ is a $\Gamma$-form for $w$, and then 4.14 yields
$\Pi^{+}(w)=[b^{-1},k,b]$ where $k=\Pi(k_{1},\cdots,k_{n})\in K$. This simply
means that $\Pi^{+}(w)=k^{b}$, since $b^{-1}=[b^{-1},\mathbb{1},\mathbb{1}]$,
$k=[\mathbb{1},k,\mathbb{1}]$, and $b=[\mathbb{1},\mathbb{1},b]$. Thus
$\Pi^{+}(w)\in\mathbb{X}$. Since $\mathbb{X}$ is closed under inversion, we
have thus shown that $\mathbb{X}$ is a partial subgroup of $\Cal{L}$.

Finally, let $c\in\Cal{L}^{+}$ be given so that
$(c^{-1},f^{\prime},c)\in\mathbb{D}^{+}$ (and where
$f^{\prime}\in\mathbb{X}$). Suppose that
$\Pi^{+}(c^{-1},f^{\prime},c)\notin\mathbb{X}$. Then
$f^{\prime}\notin\Cal{N}$, so $f^{\prime}\notin\Cal{L}$ by (4), and then
$f^{\prime}=\Pi^{+}(g^{-1},f,g)$ for some $f\in\Cal{N}$ and some
$g\in\Cal{L}^{+}$, and where $S_{f^{\prime}}=S_{(g^{-1},f,g)}$. Set
$u=(c^{-1},g^{-1},f,g,c)$. Then $u\in\mathbb{D}^{+}$ via
$(S_{f^{\prime}})^{c}$, and $\Pi^{+}(u)=\Pi^{+}(c^{-1}g^{-1},f,gc)$. Thus
$\Pi^{+}(c^{-1},f^{\prime},c)\in\mathbb{X}$ after all, and
$\mathbb{X}\trianglelefteq\Cal{L}^{+}$. Thus $\mathbb{X}=\Cal{N}$, and the
proof is complete. ∎

Let $\overline{L}$ be the quotient locality $\Cal{L}/\Cal{N}$, let
$\rho:\Cal{L}\to\overline{\Cal{L}}$ be the quotient map, and let $\rho^{*}$ be
the induced homomorphism
$\mathbb{W}(\Cal{L})\to\mathbb{W}(\overline{\Cal{L}})$ of free monoids. For
any subset or element $X$ of $\Cal{L}$, $\overline{X}$ shall denote the image
of $X$ under $\rho$. We extend this convention to subsets and elements of
$\mathbb{W}(\overline{\Cal{L}})$ in the obvious way. Set
$\overline{\Delta}=\\{\overline{P}\mid P\in\Delta\\}$.

###### Lemma 5.6

Assume that $T\nleq R$, and set
$\overline{F}=\Cal{F}_{\overline{S}}(\overline{\Cal{L}})$. Then
$\overline{R}^{\overline{\Cal{F}}}\subseteq\overline{\Delta}$.

###### Demonstration Proof

Let $U\in R^{\Cal{F}}$. As $T\nleq R$ we then have $T\nleq U$ by I.3.1(a).
Then $U$ is a proper subgroup of $UT$, so $UT\in\Delta$ by 4.2(1). Then
$\overline{U}=\overline{UT}\in\overline{\Delta}$. ∎

###### Lemma 5.7

Assume that $T\nleq R$. There is then a homomorphism
$\sigma:\Cal{L}^{+}\to\overline{\Cal{L}}$ such that the restriction of
$\sigma$ to $\Cal{L}$ is the quotient map $\rho$.

###### Demonstration Proof

Set $\overline{\mathbb{D}}=\mathbb{D}(\overline{\Cal{L}})$ and let
$\overline{\Pi}:\overline{\mathbb{D}}\to\overline{\Cal{L}}$ be the product in
$\overline{\Cal{L}}$. As
$\overline{R}^{\overline{\Cal{F}}}\subseteq\overline{\Delta}$ by 5.6, we have
$\overline{\Phi}\subseteq\overline{\mathbb{D}}$, and so there is a mapping
$\lambda:\Phi\to\overline{\Cal{L}}$ given by
$\phi\lambda=\overline{\Pi}(\phi\rho^{*})$, where
$\rho^{*}:\mathbb{W}(\Cal{L})\to\mathbb{W}(\overline{\Cal{L}})$ is the
homomorphism of free monoids induced by $\rho$. That is:
$\phi\lambda=\overline{\Pi}(\overline{\phi})$.

Let $\phi_{1},\phi_{2}\in\Phi$ with $\phi_{1}\sim\phi_{2}$, and write
$\phi_{i}=(x_{i}^{-1},h_{i},y_{i})$. Then
$(x_{2}x_{1}^{-1})h_{1}=h_{2}(y_{2}y_{1}^{-1})$ (in $M$), and so

$(\overline{x}_{2}\overline{x}_{1}^{-1})\overline{h}_{1}=\overline{h}_{2}(\overline{y}_{2}\overline{y}_{1}^{-1})$
$None$

in $\overline{M}$. As $(x_{2},x_{1}^{-1},h_{1})$ and
$(h_{2},y_{2},y_{1}^{-1})$ are in $\mathbb{D}^{+}$ via the appropriate
conjugates of $R$, we have
$(\overline{x}_{2},\overline{x}_{1}^{-1},\overline{h}_{1})$ and
$(\overline{h}_{2},\overline{y}_{2},\overline{y}_{1}^{-1})$ in
$\overline{\mathbb{D}}$, and then

$\overline{\Pi}(\overline{x}_{2},\overline{x}_{1}^{-1},\overline{h}_{1})=\overline{\Pi}(\overline{h}_{2},\overline{y}_{2},\overline{y}_{1}^{-1})$

by (1) and $\overline{\mathbb{D}}$-associativity. A standard cancellation
argument (cf. I.2.4(a)) then yields
$\overline{\Pi}(\overline{\phi}_{1})=\overline{\Pi}(\overline{\phi}_{2})$, and
thus $\lambda$ is constant on $\sim$-classes.

Now suppose that $\phi\in\Phi_{g}$ for some $g\in\Cal{L}$. That is, suppose
that $\phi\in\Phi\cap\mathbb{D}$ and that $g=\Pi(\phi)$. Then
$\phi\lambda=\overline{g}$ since $\rho$ is a homomorphism of partial groups,
and so $\lambda$ is constant on $\approx$-classes. This shows that there is a
(well-defined) mapping $\sigma:\Cal{L}^{+}\to\overline{\Cal{L}}$ given by
$\rho$ on $\Cal{L}$ and by $\lambda$ on $\Cal{L}_{0}^{+}$. It remains only to
check that $\sigma$ is a homomorphism.

Let $w=(f_{1},\cdots,f_{n})\in\mathbb{D}^{+}$. If $w\in\mathbb{D}$ then
$w\sigma^{*}=w\rho^{*}\in\overline{\mathbb{D}}$ and
$\overline{\Pi}(w\sigma^{*})=(\Pi(w))\sigma$. On the other hand, suppose that
$w\notin\mathbb{D}$. Then $f_{i}=[\phi_{i}]$ for some $\phi_{i}\in\Phi$, and
there is a $\Gamma$-form $\gamma=(\phi_{1},\cdots,\phi_{n})$ of $w$. Thus the
word $w_{\gamma}=\phi_{1}\circ\cdots\circ\phi_{n}$ has the property that
$S_{w_{\gamma}}$ is an $\Cal{F}$-conjugate of $R$, and hence
$\overline{w_{\gamma}}\in\overline{\mathbb{D}}$. Then

$\overline{\Pi}((w_{\gamma})\sigma^{*})=\overline{\Pi}(\overline{w_{\gamma}}).$

Write $\phi_{i}=(x_{i}^{-1},h_{i},y_{i})$. Then

$(\Pi^{+}(w_{\gamma}))\sigma=[x_{1}^{-1},\Pi(w_{0}),y_{n}]\sigma$

where $w_{0}\in\mathbb{D}(M)$ is given by the formula in 4.14(b). One then
obtains $\overline{\Pi}((w_{\gamma})\sigma^{*})=(\Pi^{+}(w_{\gamma}))\sigma$
via $\overline{\mathbb{D}}$-associativity, and so $\sigma$ is a homomorphism,
as desired. ∎

###### Lemma 5.8

$\Cal{N}^{*}\cap\Cal{L}=\Cal{N}$.

###### Demonstration Proof

Let $\sigma:\Cal{L}^{+}\to\overline{\Cal{L}}$ be a homomorphism, as in 5.7,
whose restriction to $\Cal{L}$ is the quotient map
$\rho:\Cal{L}\to\overline{\Cal{L}}$. Then $Ker(\sigma)\cap\Cal{L}=Ker(\rho)$,
where $Ker(\rho)=\Cal{N}$, by I.5.3. As
$Ker(\sigma)\trianglelefteq\Cal{L}^{+}$, by I.1.14, the lemma follows. ∎

Theorem 5.1 now follows from lemmas 5.3, 5.5, and 5.8.

Section 6: Proper localities

Recall from I.3.6 that if $G$ is a group and $p$ is a prime then $Syl_{p}(G)$
is the (possibly empty) set of maximal $p$-subgroups $S$ of $G$ such that
every $p$-subgroup of $G$ is conjugate in $G$ to a subgroup of $S$. For any
group $G$ with $Syl_{p}(G)\neq\emptyset$, define $O_{p}(G)$ to be the subgroup
$\bigcap Syl_{p}(G)$ of $G$. Thus $O_{p}(G)$ is the largest normal
$p$-subgroup of $G$. If $(\Cal{L},\Delta,S)$ is a locality then there is
another usage, whereby $O_{p}(\Cal{L})=\mathbb{1}^{\star}$, where
$(\Omega,\star)$ is the stratification on $\Cal{F}_{S}(\Cal{L})$ induced from
$\Cal{L}$. If it so happens that $\Cal{L}$ is a group then these two
definitions of $O_{p}(\Cal{L})$ agree by I.3.7, so there is no cause for
confusion.

###### Definition Definition 6.1

A group $G$ is of characteristic $p$ if $Syl_{p}(G)\neq\emptyset$ and
$C_{G}(O_{p}(G))\leq O_{p}(G)$.

For a fusion system $\Cal{F}$ on a group $S$ there is a largest subgroup $X$
of $S$ such that $\Cal{F}=N_{\Cal{F}}(X)$, by 1.9. In the case that $S$ is a
$p$-group we write $O_{p}(\Cal{F})$ for $X$.

###### Lemma 6.2

Let $(\Cal{L},\Delta,S)$ be a locality on $\Cal{F}=\Cal{F}_{S}(\Cal{L})$, and
assume that $N_{\Cal{L}}(P)$ is of characteristic $p$ for all $P\in\Delta$.
Let $R\leq S$ be a subgroup of $S$. Then $R\trianglelefteq\Cal{L}$ if and only
if $R\trianglelefteq\Cal{F}$. In particular, we have
$O_{p}(\Cal{F})=O_{p}(\Cal{L})$.

###### Demonstration Proof

As $\Cal{F}$ is generated by the conjugation maps $c_{g}:S_{g}\to S$ for
$g\in\Cal{L}$, we have $R\trianglelefteq\Cal{F}$ if $R\trianglelefteq\Cal{L}$.
So assume $R\trianglelefteq\Cal{F}$ and, by way of contradiction, that $R$ is
not normal in $\Cal{L}$. Among all elements of $\Cal{L}$ not in
$N_{\Cal{L}}(R)$, choose $g$ so that $dim(S_{g})$ is as large as possible.
Then $R\nleq S_{g}$, since $c_{g}:S_{g}\to S$ is an $\Cal{F}$-homomorphism. In
particular, $S_{g}\neq S$. As $S_{g}\in\Omega$, $S_{g}$ is then a proper
subgroup of $N_{S}(S_{g})$ by I.3.2.

Set $P=S_{g}$, $P^{\prime}=P^{g}$, and let $Q\in P^{\Cal{F}}$ be fully
normalized in $\Cal{F}$. By by 3.2(a) there exists $x\in\Cal{L}$ such that
$P^{x}=Q$ and such that $N_{S}(P)\leq S_{x}$. As $P\in\Omega$ we have
$dim(P)<dim(N_{S}(P))$ by I.3.2, and so $R\leq S_{x}$. Since
$Q\in(P^{\prime})^{\Cal{F}}$ there exists also $y\in\Cal{L}$ such that
$(P^{\prime})^{y}=Q$ and such that $N_{S}(P^{\prime})R\leq S_{y}$.

Note that $(x^{-1},g,y)\in\mathbb{D}$ via $Q$, and that $f:=\Pi(x^{-1},g,y)\in
N_{\Cal{L}}(Q)$. Note also that $(x,x^{-1},g,y,y^{-1})\in\mathbb{D}$ via $P$,
and that

$\Pi(x,f,y^{-1})=\Pi(x,x^{-1},g,y,y^{-1})=\Pi(g)$

by $\mathbb{D}$-associativity (I.1.4). If $R\leq S_{f}$ then $R\leq
S_{(x,f,y^{-1})}$, and then $R\leq S_{g}$. Thus $R\nleq S_{f}$, and we may
therefore replace $g$ with $f$, and $P$ with $Q$. That is, we may assume that
$P$ is fully normalized in $\Cal{F}$ and that $g$ is in $N_{\Cal{L}}(P)$.

Set $H=N_{\Cal{L}}(P)$, set $D=N_{R}(P)$, and set
$\Cal{E}=\Cal{F}_{N_{S}(P)}(H)$. Then $D\trianglelefteq\Cal{E}$, and so the
conjugation map $c_{g}:P\to P$ extends to an $\Cal{E}$-automorphism of $PD$.
Thus, there exists $h\in N_{H}(PD)$ such that $c_{h}:PD\to PD$ restricts to
$c_{g}$ on $P$. Then $gh^{-1}\in C_{H}(P)$. Since $H$ is is of characteristic
$p$ by hypothesis, we then have $gh^{-1}\in P$. Then $gh^{-1}\in N_{H}(D)$.
Since $c_{h}$ extends to an $\Cal{F}$-homomorphism which fixes $R$, we have
$D^{h}=D$, and so also $D^{g}=D$. Thus $D\leq P$, and then $R\leq P$ by I.3.2.
This contradicts the choice of $g$, and completes the proof. ∎

By a $p^{\prime}$-group we mean a torsion group $G$ all of whose elements have
order relatively prime to $p$. In I.3.2 it was seen that the finite-
dimensionality condition on a locality $(\Cal{L},\Delta,S)$ has an important
consequence for the structure of the $p$-group $S$. Namely, it was shown that
if $X\in\Omega$ and $Y\leq S$ is a subgroup of $S$ with $X<Y$ (proper
inclusion), then $X<N_{Y}(X)$. In order to begin to analyze the interplay
between $p$-subgroups and $p^{\prime}$-subgroups of $\Cal{L}$, we shall need
to generalize Thompson’s $A\times B$-Lemma (concerning finite groups). First,
we recall the definition I.6.2: A $p$-group $S$ has the normalizer-increasing
property if, for every pair $X$ and $Y$ of subgroups of $S$ with $X<Y$, we
have $X<N_{Y}(X)$. (The remark following I.6.2 provides an example of a
countable, locally finite $p$-group which does not have the normalizer-
increasing property.

###### Lemma 6.3

Let $G$ be a torsion group, and let $A$ and $B$ be subgroups of $G$ such that
the order of every element of $A$ is relatively prime to the order of every
element of $B$. Suppose that $[A,B]\leq C_{B}(A)$. Then $[A,B]=\mathbb{1}$.

###### Demonstration Proof

Let $a\in A$ and $b\in B$. Then $a^{-1}a^{b}=[a,b]$ commutes with $a$, by
hypothesis, and so $a^{b}$ commutes with $\langle a\rangle$. As $|a|=|a^{b}|$
is relatively prime to the order of any element of $B$, the same is then true
of $a^{-1}a^{b}$. As $[a,b]\in B$ by hypothesis, we conclude that $[a,b]=1$,
and thus $[A,B]=\mathbb{1}$. ∎

###### Lemma 6.4 (Thompson's ``$A\timesB$" Lemma)

Let $G$ be a torsion group, and assume:

Then $G$ is isomorphic to the direct product $A\times P$.

###### Demonstration Proof

As $P\cap A=\mathbb{1}$ it suffices to show that $[P,A]=1$. Suppose false, so
that $C_{P}(A)$ is a proper subgroup of $P$. Set $Q=N_{P}(C_{P}(A))$. $B\leq
C_{P}(A)\trianglelefteq Q$. Thus $[Q,B]\leq C_{P}(A)$; and thus $[[Q,B],A]=1$.
Also $[[B,A],Q]=1$ since $B\leq C_{P}(A)$, and then the Three Subgroups Lemma
yields $[[A,Q],B]=1$. That is, we have $[A,Q]\leq C_{Q}(B)$, and hence
$[A,Q]\leq C_{Q}(A)$ by (ii). Then $Q\leq C_{P}(A)$ by 6.3. The normalizer-
increasing hypothesis for $P$ then yields $C_{P}(A)=P$, as required. ∎

By a $p$-local subgroup of a group $G$ we mean a subgroup of the form
$N_{G}(P)$, where $P$ is a $p$-subgroup of $G$.

###### Lemma 6.5

Let $G$ be a countable, locally finite group of characteristic $p$.

###### Demonstration Proof

Let $K\trianglelefteq G$ be a normal subgroup of $G$ and set $R=O_{p}(K)$.
Then $[O_{p}(G),K]\leq R$, and so $[O_{p}(G),C_{K}(R),C_{K}(R)]=\mathbb{1}$.
Then the group $O^{p}(C_{K}(R))$ generated by the $p^{\prime}$-elements of
$C_{K}(R)$ is contained in $C_{G}(O_{p}(G))$ by 6.3, and thus
$O^{p}(C_{K}(R))\leq O_{p}(G)$. Then $C_{K}(R)$ is a normal $p$-subgroup of
$K$, and so $C_{K}(R)\leq R$. Thus (a) holds.

Next, let $U$ be a $p$-subgroup of $G$, and set $H=N_{G}(U)$, $P=O_{p}(H)$,
and $Q=O_{p}(G)$. Then $N_{Q}(U)\leq P$. Let $A$ be a $p^{\prime}$-subgroup of
$C_{H}(P)$. Then $[N_{Q}(U),A]=\mathbb{1}$. Assuming now that $Q$ has the
normalizer-increasing property, it follows from 6.4 that $[Q,A]=\mathbb{1}$.
Then $A\leq Q$, and thus $A=\mathbb{1}$, proving (b).

For the proof of (c), notice that $C_{G}(V)\trianglelefteq G$, and that $X$ is
the intersection of $C_{G}(V)$ with the preimage in $G$ of the normal subgroup
$C_{G/V}(O_{p}(G)/V)$ of $G/V$. Thus $X\trianglelefteq G$. Each
$p^{\prime}$-subgroup of $X$ centralizes $O_{p}(G)$ by 6.3, so $X$ is a
$p$-group. ∎

Recall that, for any fusion system $\Cal{F}$ on $S$, $\Cal{F}^{c}$ is the set
of $\Cal{F}$-centric subgroups of $S$; i.e. subgroups $P$ of $S$ such that
$C_{S}(Q)\leq Q$ for all $Q\in P^{\Cal{F}}$.

###### Definition Definition 6.6

Let $(\Cal{F},\Omega,\star)$ be a stratified fusion system on the countable,
locally finite $p$-group $S$. A subgroup $P\leq S$ is:

Denote by $\Cal{F}^{cr}$ the set of all subgroups $P\leq S$ such that $P$ is
both $\Cal{F}$-centric and $\Cal{F}$-radical, by $\Cal{F}^{q}$ the set of all
subgroups $P\leq S$ such that $P$ is $\Cal{F}$-quasicentric, and by
$\Cal{F}^{s}$ the set of all subgroups $P\leq S$ such that $P$ is
$\Cal{F}$-subcentric.

###### Definition Remark

The definition given above of $\Cal{F}$-radical subgroup differs from the
standard one (in the case where $S$ is finite). Namely, in the standard
definition a subgroup $P$ of $S$ is $\Cal{F}$-radical if
$Inn(P)=O_{p}(Aut_{\Cal{F}}(P))$, but we find the definition given above to be
more useful. We mention also that the notion of $\Cal{F}$-subcentric subgroup
was first explored by Ellen Henke [He2].

###### Definition Definition 6.7

Let $(\Cal{L},\Delta,S)$ be a locality. Then $\Cal{L}$ is proper if:

###### Lemma 6.8

Let $(\Cal{L},\Delta,S)$ be a locality on $\Cal{F}$. Assume

Then there exists a partial normal subgroup $\Theta\trianglelefteq\Cal{L}$
such that $S\cap\Theta=\mathbb{1}$ and such that, upon identifying $S$ with
its image in $\Cal{L}/\Theta$, the quotient $\Cal{L}/\Theta$ is a proper
locality on $\Cal{F}$.

###### Demonstration Proof

As $\Delta\subseteq\Cal{F}^{c}$ it follows from I.6.1 that for each
$P\in\Delta$ there is a largest normal $p^{\prime}$-subgroup $\Theta(P)$
contained in $N_{\Cal{L}}(P)$, and that then $N_{\Cal{L}}(P)/\Theta(P)$ is of
characteristic $p$. Define $\Theta$ to be the union of the groups $\Theta(P)$
for $P\in\Delta$. With (1) the hypotheses of I.6.3 are fulfilled, and we
obtain $\Theta\trianglelefteq\Cal{L}$, $\Cal{F}=\Cal{F}_{S}(\Cal{L}/\Theta)$,
and $\Cal{L}/\Theta$ is a proper locality on $\Cal{F}$. ∎

###### Lemma 6.9

Let $(\Cal{L},\Delta,S)$ be a proper locality on $\Cal{F}$. Then
$\Delta\subseteq\Cal{F}^{s}$, and for any $P\in\Delta$:

###### Demonstration Proof

Let $P\in\Delta$, set $M=N_{\Cal{L}}(P)$, and let $Q\in P^{\Cal{F}}$ with $Q$
fully normalized in $\Cal{F}$. Then $Q=P^{g}$ for some $g\in\Cal{L}$ by I.3.5,
and then the conjugation map $c_{g}:M\to M^{g}$ is an isomorphism by I.2.3(b).
As $\Cal{F}^{cr}$, $\Cal{F}^{c}$, $\Cal{F}^{q}$, and $\Cal{F}^{s}$ are
$\Cal{F}$-invariant by 1.9, it suffices to establish that $P\in\Cal{F}^{s}$
and that (a) through (c) hold, under the assumption (which we now make) that
$P$ itself is fully normalized in $\Cal{F}$.

Regard $M$ as a locality whose set of objects is the set of all overgroups of
$O_{p}(M)$ in $N_{S}(P)$. Since the group $M$ is of characteristic $p$ by
(PL2), the locality $M$ is proper as a consequence of 6.6(b). As
$N_{\Cal{F}}(P)=\Cal{F}_{N_{S}(P)}(M)$ by 3.2(b), 6.2 implies that
$O_{p}(N_{\Cal{F}}(P))=O_{p}(M)$. Here $C_{S}(O_{p}(M))\leq O_{p}(M)$ (as $M$
is of characteristic $p$), so 3.4(a) yields $O_{p}(M)\in\Cal{F}^{c}$. Thus
$P\in\Cal{F}^{s}$, and we have shown moreover that if $P\in\Cal{F}^{cr}$ then
$P=O_{p}(M)$. On the other hand, if $P=O_{p}(M)$ then
$P=O_{p}(N_{\Cal{F}}(P))$, and so (a) holds.

Set $K=C_{\Cal{L}}(P)$. Then $C_{\Cal{F}}(P)=\Cal{F}_{C_{S}(P)}(K)$ by 3.2(b).
If $P\in\Cal{F}^{c}$ then $C_{S}(P)=Z(P)\leq Z(K)$, and then $O^{p}(K)$ is a
normal $p^{\prime}$-subgroup of $K$ by I.6.1. Since $K$ is of characteristic
$p$ by 6.6(a), $K$ has no non-identity normal $p^{\prime}$-subgroups, so we
conclude that $O^{p}(K)=1$. Thus $K=Z(P)$. Conversely, if $K=Z(P)$ then
$C_{S}(P)\leq P$. As $P$ is fully normalized in $\Cal{F}$, 3.4(a) then yields
$P\in\Cal{F}^{c}$. This establishes (b).

Suppose next that $P\in\Cal{F}^{q}$, so that $C_{\Cal{F}}(P)$ is the trivial
fusion system on $C_{S}(P)$. Then $N_{K}(U)/C_{K}(U)$ is a $p$-group for every
subgroup $U$ of $C_{S}(P)$. Take $U=O_{p}(K)$. Thus $K/Z(U)$ is a $p$-group,
so $K$ is a $p$-group, and then $K=C_{S}(P)$ is a normal $p$-subgroup of $M$.
Conversely, if $K\leq O_{p}(M)$ then $C_{\Cal{F}}(P)$ is trivial, so we have
(c). ∎

Section 7: Theorem A

This section concerns expansions of proper localities and of homomorphic
images of proper localities. Throughout, $(\Cal{L},\Delta,S)$ will be a proper
locality on $\Cal{F}$, $(\Omega,\star)$ will be the stratification induced
from $\Cal{L}$, and the assertion that a subgroup $X$ of $S$ is “fully
normalized in $\Cal{F}$” will mean that $X$ is fully normalized with respect
to $(\Omega,\star)$. We shall see somewhat later (in 8.3) that $\Cal{F}$ is
inductive, so by 2.9 the notion of being fully normalized in $\Cal{F}$ is
independent of the given stratification.

###### Lemma 7.1

Let $(\Cal{L},\Delta,S)$ be a proper locality on $\Cal{F}$ and let
$R\in\Cal{F}^{s}$. Assume:

Set $M=N_{\Cal{L}}(R)$. Then:

In particular, if $R\in\Omega_{S}(\Cal{L})$ then $\Cal{L}$ and $R$ satisfy the
conditions of Hypothesis 4.2.

###### Demonstration Proof

We begin with the observation that the normalizer in $\Cal{L}$ of any subgroup
of $S$ is a partial subgroup of $\Cal{L}$. Set $\Delta_{R}=\\{P\in\Delta\mid
R\trianglelefteq P\\}$. We will show that $(M,\Delta_{R},N_{S}(R))$ is a
proper locality on $N_{\Cal{F}}(R)$.

Let $Q\in\Delta$ with $R\leq Q$. Then either $R=Q$ or $R<N_{Q}(R)$, and in
either case (2) implies that $N_{Q}(R)\in\Delta$. Thus:

Set $\Cal{F}_{R}=\Cal{F}_{N_{S}(R)}(M)$. Then $\Delta_{R}$ is
$\Cal{F}_{R}$-closed since $\Delta_{R}$ is closed with respect to overgroups
in $N_{S}(R)$, and since $\Delta_{R}$ is closed with respect to conjugation
maps $c_{g}:P\to P^{g}$ with $g\in M$ and with $P\leq S_{g}$. Let
$w\in\mathbb{W}(M)\cap\mathbb{D}(\Cal{L})$, and set $Q=S_{w}$. Then (3) yields
$w\in\mathbb{D}(\Cal{L})$ via $N_{Q}(R)$, and we have thus verified that
$(M,\Delta_{R})$ is an objective partial group (I.2.1), and that
$(M,\Delta_{R},N_{S}(R))$ is a pre-locality. By I.2.15, $\Omega_{N_{S}(R)}(M)$
is finite-dimensional.

Set $V=N_{S}(R)$. An argument that should by now be familiar will show that
$V$ is a maximal $p$-subgroup of $M$. Namely, let $V\leq X$ where $X$ is a
$p$-subgroup of $M$. By I.3.8 there exists $g\in\Cal{L}$ with $X^{g}\leq S$.
Then $dim(V^{g})=dim(X^{g})$ since $R$ is fully normalized, and so
$(V^{g})^{*}=(X^{g})^{*}$. Since $V=N_{V^{*}}(R)$ we get
$V^{g}=N_{(V^{g})^{*}}(R^{g})$, so $V^{g}\geq X^{g}$, and $V=X$. Thus
$N_{S}(R)$ is a maximal $p$-subgroup of $M$, and $(M,\Delta_{R},V)$ is a
locality on $\Cal{F}_{R}$.

For any $P\in\Delta_{R}$ we have

$N_{\Cal{L}_{R}}(P)=N_{N_{\Cal{L}}(P)}(R),$

and 6.5(b) then implies that $N_{\Cal{L}_{R}}(P)$ is of characteristic $p$.
Thus $\Cal{L}_{R}$ satisfies the condition (PL2) in the definition 6.7 of
proper locality.

If $R\in\Delta$ then $M$ is a subgroup of $\Cal{L}$ of characteristic $p$
(since $\Cal{L}$ is proper), and then $N_{\Cal{F}}(R)=\Cal{F}_{R}$ by 3.2.
Thus (a) and (b) hold in this case, and we may assume that $R\notin\Delta$.
Set $Q=O_{p}(N_{\Cal{F}}(R))$ and suppose next that $R=Q$. Then
$R\in\Cal{F}^{c}$ by (1), and so $R\in\Cal{F}^{cr}$ (cf. definition 6.6).
Since $\Cal{F}^{cr}\subseteq\Delta$, and since we have eliminated the case
where $R\in\Delta$, we conclude that $R\neq Q$. As $\Cal{F}_{R}$ is a
subsystem of $N_{\Cal{F}}(R)$ we have $Q\leq O_{p}(\Cal{F}_{R})$, so
$R<O_{p}(\Cal{F}_{R})$. Thus $O_{p}(\Cal{F}_{R})\in\Delta$, and so
$(\Cal{F}_{R})^{cr}\subseteq\Delta_{R}$. That is: $\Cal{L}_{R}$ satisfies the
condition (PL1) in definition 6.7, Condition (PL3) (the normalizer-increasing
property) is inherited by $N_{S}(R)$ from $S$, and thus
$(\Cal{L}_{R},\Delta_{R},N_{S}(R))$ is a proper locality. Then
$O_{p}(\Cal{F}_{R})=O_{p}(M)$ by 6.2, and then $M$ is a group since
$O_{p}(\Cal{F}_{R})\in\Delta$. As $Q\leq O_{p}(\Cal{F}_{R})$, (1) implies that
$Q$ is centric in $\Cal{F}$, hence also in $\Cal{F}_{R}$, and so $M$ is of
characteristic $p$. Thus (a) holds.

As $N_{S}(R)$ is a maximal $p$-subgroup of $M$, I.3.11 yields $N_{S}(R)\in
Syl_{p}(M)$. Since $Q\in\Delta$, each $N_{\Cal{F}}(R)$-isomorphism is the
restriction of a conjugation map $c_{g}:X\to Y$ with $Q\leq X\cap Y$, and for
some $g\in L$ such that $R^{g}=R$. Thus $N_{\Cal{F}}(R)$-isomorphisms are
$\Cal{F}_{R}$-isomorphisms, and this completes the proof of (b). ∎

Our aim now is to extend the results of sections 4 and 5 beyond elementary
expansions. The arguments amount to an exercise with Zorn’s Lemma, and will be
based on the restricted notion of expansion in definition 4.1 (where the
homomorphism $\eta$ is assumed to be an inclusion map).

###### Theorem 7.2

Let $(\Cal{L},\Delta,S)$ be a proper locality on $\Cal{F}$ and let
$\Delta^{+}$ be an $\Cal{F}$-closed collection of subgroups of $S$ such that
$\Delta\subseteq\Delta^{+}\subseteq\Cal{F}^{s}$. Then the following hold.

###### Demonstration Proof

Let $\widetilde{\mathbb{E}}$ be the class of all expansions
$(\Cal{L}^{\prime},\Delta^{\prime},S)$ of $\Cal{L}$ such that $\Cal{L}$ is a
proper locality, with $\Cal{F}_{S}(\Cal{L}^{\prime})=\Cal{F}$, and such that
$\Cal{L}^{+}$ is generated by $\Cal{L}$ as a partial group. For
$\Cal{L}_{1},\Cal{L}_{2}\in\widetilde{\mathbb{E}}$, we say that a homomorphism
$\phi:\Cal{L}_{1}\to\Cal{L}_{2}$ is rigid if $\phi$ restricts to the identity
map on $\Cal{L}$. Let $\approx$ be the equivalence relation on
$\widetilde{\mathbb{E}}$ given by $\Cal{L}_{1}\approx\Cal{L}_{2}$ if there
exists a rigid isomorphism $\Cal{L}_{1}\to\Cal{L}_{2}$.

For any locality $(\Cal{L},\Delta,S)$, each of the groups $N_{\Cal{L}}(P)$ for
$P\in\Delta$ is countable. Then $S$ is countable, and so the cardinality of
$\Delta$ is at most $\aleph_{1}$. For any $P,Q\in\Delta$ the set
$N_{\Cal{L}}(P,Q)$ is countable, and since $\Cal{L}$ is the union of the sets
$N_{\Cal{L}}(P,Q)$, the cardinality of $\Cal{L}$, and then also of
$\mathbb{W}(\Cal{L})$ and $\mathbb{D}(\Cal{L})$, is at most $\aleph_{1}$.
There is then a set of isomorphism classes of localities, and a set
$\mathbb{E}$ of $\approx$-classes of expansions
$\Cal{L}^{\prime}\in\widetilde{\mathbb{E}}$ of $\Cal{L}$. Further, we may view
$\mathbb{E}$ as a poset, whereby $X_{1}\preceq X_{2}$ if there exist
representatives $\Cal{L}_{1}\in X_{1}$ and $\Cal{L}_{2}\in X_{2}$ such that
there exists a rigid injective homomorphism $\phi\Cal{L}_{1}\to\Cal{L}_{2}$.
Since $\Cal{L}_{1}$ and $\Cal{L}_{2}$ are generated by $\Cal{L}$, $\phi$ is
then the unique rigid homomorphism $\Cal{L}_{1}\to\Cal{L}_{2}$.

Let $\\{X_{i}\\}_{i\in I}$ be a totally ordered subset of $\mathbb{E}$, and
choose $\Cal{L}_{i}\in X_{i}$ for all $i$. Then $\\{\Cal{L}_{i}\\}_{i\in I}$
comes equipped with rigid homomorphisms
$\phi_{i,j}:\Cal{L}_{i}\to\Cal{L}_{j}$, and the uniqueness of $\phi_{i,j}$
implies that $\\{\Cal{L}_{i}\\}_{i\in I}$ is a directed system in this way.
The category of partial groups possesses limits (Appendix B in Part I), so the
direct limit of this directed system is a partial group $\Cal{L}^{\prime}$.
For convenience, we may then identify each $\Cal{L}_{i}$ with its image in
$\Cal{L}^{\prime}$, and take the homomorphisms $\phi_{i,j}$ to be inclusion
maps.

Let $\Delta^{\prime}$ be the union of the sets $\Delta_{i}$. Since each
$(\Cal{L}_{i},\Delta_{i},S)$ is a locality on $\Cal{F}$, $\Delta^{\prime}$ is
$\Cal{F}$-closed. For any $w\in\mathbb{D}(\Cal{L}^{\prime})$ we have
$w\in\mathbb{D}(\Cal{L}_{i})$ for some $i$, and then
$(\Cal{L}^{\prime},\Delta^{\prime})$ is an objective partial group, and
$(\Cal{L}^{\prime},\Delta^{\prime},S)$ is a pre-locality. Since the
stratification on $\Cal{F}$ induced from $\Cal{L}_{i}$ is the same as that
which is induced from $\Cal{L}$, $\Omega_{S}(\Cal{L}^{\prime})$ is finite-
dimensional. Evidently $S$ is a maximal $p$-subgroup of $\Cal{L}^{\prime}$, so
$(\Cal{L}^{\prime},\Delta^{\prime},S)$ is a locality. For any
$P\in\Delta^{\prime}$ we have $P\in\Delta_{i}$ for some $i$, and then
$N_{\Cal{L}^{\prime}}(P)=N_{\Cal{L}_{i}}(P)$ is of characteristic $p$. Thus
$\Cal{L}^{\prime}$ is a proper locality, and $\Cal{L}^{\prime}$ is an
expansion of $\Cal{L}$. The $\approx$-class of $\Cal{L}^{\prime}$ is thus an
upper bound for $\Cal{C}$ in $\mathbb{E}$, and so $\mathbb{E}$ contains a
maximal element $Y$ by Zorn’s Lemma. Rather than introduce further notation,
let us now write $(\Cal{L}^{\prime},\Delta^{\prime},S)$ for a representative
of $Y$.

Set $\widetilde{D}=\\{P\leq S\mid P^{\star}\in\Delta\\}$. Then
$(\Cal{L},\widetilde{D},S)$ is an expansion of $\Cal{L}$ by I.3.15, and there
is then no loss of generality in assuming that
$\Delta=\widetilde{\Delta}\cap\Delta^{+}$. Suppose now that
$\Delta^{\prime}\neq\Delta^{+}$, choose $R\in\Delta^{+}-\Delta^{\prime}$ so
that $dim(R)$ is as large as possible, and set $\Delta^{*}=\Delta^{\prime}\cup
R^{\Cal{F}}$. Then $\Delta^{*}$ is $\Cal{F}$-closed. Since
$\Delta^{+}\subseteq\Cal{F}^{s}$, we may assume that $R$ has been chosen also
so that $R$ is fully normalized in $\Cal{F}$ and so that
$O_{p}(N_{\Cal{F}}(R))\in\Cal{F}^{c}$. Then 7.1 implies that the normalizer
$M=N_{\Cal{L}}(R)$ is a subgroup of $\Cal{L}$ of characteristic $p$ and that
$N_{S}(R)\in Syl_{p}(M)$. Theorem 4.3 then yields the existence of a proper
locality $(\Cal{L}^{*},\Delta^{*},S)$ on $\Cal{F}$ whose restriction to
$\Delta^{\prime}$ is $\Cal{L}^{\prime}$, with
$\Cal{F}_{S}(\Cal{L}^{*})=\Cal{F}$, and such that $\Cal{L}^{\prime}$ generates
$\Cal{L}^{*}$ as a partial group. This contradicts the maximality of $Y$, and
completes the proof. ∎

###### Theorem 7.3

Let $(\Cal{L},\Delta,S)$ be a proper locality on $\Cal{F}$, and let
$(\Cal{L}^{+},\Cal{D}^{+},S)$ be an expansion of $\Cal{L}$ to an
$\Cal{F}$-closed subset $\Delta^{+}$ of $F^{s}$. Let $\mathfrak{N}$ be the set
of partial normal subgroups of $\Cal{L}$, and let $\mathfrak{N}^{+}$ be the
set of partial normal subgroups of $\Cal{L}^{+}$. For each
$\Cal{N}\in\mathfrak{N}$ there is then a unique
$\Cal{N}^{+}\in\mathfrak{N}^{+}$ such that $\Cal{N}=\Cal{N}^{+}\cap\Cal{L}$.
In particular, we have $S\cap\Cal{N}=S\cap\Cal{N}^{+}$, and the mapping
$\Cal{N}\mapsto\Cal{N}^{+}$ is an inclusion-preserving bijection
$\mathfrak{N}\to\mathfrak{N}^{+}$.

###### Demonstration Proof

Fix $\Cal{N}\trianglelefteq\Cal{L}$ and let $\widetilde{\mathbb{E}}_{\Cal{N}}$
to be the class of expansions $(\Cal{L}^{\prime},\Delta^{\prime},S)$ of
$\Cal{L}$ such that $\Delta^{\prime}\subseteq\Delta^{+}$ and such that there
exists a unique partial normal subgroup
$\Cal{N}^{\prime}\trianglelefteq\Cal{L}^{\prime}$ with
$\Cal{L}\cap\Cal{N}^{\prime}=\Cal{N}$. As in the proof of 7.2 there is then a
set $\mathbb{E}_{\Cal{N}}$ of rigid isomorphism-classes of members of
$\widetilde{\mathbb{E}}_{\Cal{N}}$, $\mathbb{E}_{\Cal{N}}$ is a poset, and any
set of representatives for $\mathbb{E}_{\Cal{N}}$ forms a directed set. As in
the proof of 7.1, we may assume that $P\in\Delta$ whenever $P\leq S$ and
$P^{\star}\in\Delta$.

Let $\Cal{C}=\\{Y_{i}\\}_{i\in I}$ be a totally ordered subset of
$\mathbb{E}_{\Cal{N}}$, and choose representatives
$(\Cal{L}_{i},\Delta_{i},S)\in Y_{i}$. As in the proof of 7.2, these
representatives form a directed system whose direct limit is then a proper
locality $(\Cal{L}^{\prime},\Delta^{\prime},S)$ which is an expansion of
$\Cal{L}$. Also as in 7.2, we may take $\Cal{L}_{i}$ to be the restriction of
$\Cal{L}^{\prime}$ to $\Delta_{i}$, and take the connecting homomorphisms for
the directed system to be inclusion maps.

Let $\Cal{N}_{i}$ be the unique partial normal subgroup of $\Cal{L}_{i}$ whose
intersection with $\Cal{L}$ is $\Cal{N}$. and then let $\Cal{N}^{\prime}$ be
the union of the $\Cal{N}_{i}$ in $\Cal{L}^{\prime}$. Plainly,
$\Cal{N}^{\prime}$ is a partial subgroup of $\Cal{L}^{\prime}$, and is closed
under $\Cal{L}^{\prime}$-conjugation (when conjugation is defined). Thus
$\Cal{N}^{\prime}$ is a partial normal subgroup of $\Cal{L}^{\prime}$. One
observes that

$\Cal{N}^{\prime}\cap\Cal{L}=\bigcup(\Cal{N}^{\prime}\cap\Cal{L}_{i})\cap\Cal{L}=\bigcup(\Cal{N}_{i})\cap\Cal{L}=\Cal{N}.$

In order to conclude that the $\approx$-class of $\Cal{L}^{\prime}$ is an
upper bound for $\Cal{C}$ in $\mathbb{E}$ it remains to show that if $\Cal{M}$
is any partial normal subgroup of $\Cal{L}^{\prime}$ which meets $\Cal{L}$ in
$\Cal{N}$ then $\Cal{M}=\Cal{N}^{\prime}$. Indeed:

$\Cal{M}=\bigcup(\Cal{M}\cap\Cal{L}_{i})=\bigcup(\Cal{N}^{\prime}\cap\Cal{L}_{i})=\Cal{N}^{\prime}.$

Thus $\Cal{C}$ has an upper bound, and Zorn’s Lemma implies that $\mathbb{E}$
contains a maximal element $Y$.

Continuing as in the proof of 7.2, let $(\Cal{L}^{\prime},\Delta^{\prime},S)$
now denote a representative of $Y$. If $\Delta^{\prime}\neq\Delta^{+}$ then
there is an elementary expansion $(\Cal{L}^{*},\Delta^{*},S)$ of
$\Cal{L}^{\prime}$ which is not in $\widetilde{\mathbb{E}}_{\Cal{N}}$, and
then Theorem 5.1 yields a contradiction. ∎

The preceding two results are the Theorems A1 and A2, as stated in the
introduction, and we think of them as comprising the two parts of a single
Theorem A.

Recall from I.6.8 that the product of any non-empty set of partial normal
subgroups of a locality is again a partial normal subgroup. That is: if
$\mathfrak{N}\neq\emptyset$ is a set of partial normal subgroups of $\Cal{L}$,
then the partial subgroup $\Cal{M}=\langle\bigcup\mathfrak{N}\rangle$ of
$\Cal{L}$ is normal in $\Cal{L}$, and for each total ordering $\preceq$ of
$\mathfrak{N}$ and each element $x\in\Cal{M}$, there exists a finite sequence
$(\Cal{N}_{1},\cdots,\Cal{N}_{k})$ of elements of $\mathfrak{N}$, and elements
$x_{i}\in\Cal{N}_{i}$ $(1\leq i\leq k)$, such that
$\Cal{N}_{1}\preceq\cdots\preceq\Cal{N}_{k}$,
$(x_{1},\cdots,x_{k})\in\mathbb{D}$, and $x=Pi(x_{1},\cdots,x_{k})$.

###### Corollary 7.4

Let $\mathfrak{N}$ be a non-empty set of partial normal subgroups of the
proper locality $(\Cal{L},\Delta,S)$, and let $(\Cal{L}^{+},\Delta^{+},S)$ be
a proper expansion of $\Cal{L}$. Let $\Cal{M}$ be the partial subgroup of
$\Cal{L}$ generated by $\bigcup\mathfrak{N}$, and for any
$\Cal{N}\trianglelefteq\Cal{L}$ let $\Cal{N}^{+}$ be the unique partial normal
subgroup of $\Cal{L}^{+}$ such that $\Cal{N}^{+}\cap\Cal{L}=\Cal{N}$. Then
$\Cal{M}^{+}$ is generated by
$\bigcup\\{\Cal{N}^{+}\\}_{\Cal{N}\in\mathfrak{N}}$, and if
$\bigcap\mathfrak{N}\leq S$ then
$(\bigcap\mathfrak{N})^{+}=\bigcap\mathfrak{N}$.

###### Demonstration Proof

We provide the details only for the case that $\mathfrak{N}$ consists of a
pair $\\{\Cal{N}_{1},\Cal{N}_{2}\\}$ of partial normal subgroups of $\Cal{L}$
(this being the only case that will be required for later applications), and
leave the general case to the reader who is interested in carrying out the
exercise. For any pair $X_{1}$ and $X_{2}$ of non-empty subsets of $\Cal{L}$
it follows from I.1.9 that $\langle X_{1}\rangle\langle X_{2}\rangle$ is a
subset of $\langle X_{1}X_{2}\rangle$. Then
$\langle\Cal{N}_{1}^{\Cal{L}^{+}}>\langle\Cal{N}_{2}^{\Cal{L}^{+}}\rangle\subseteq\langle\Cal{N}_{1}^{\Cal{L}^{+}}\Cal{N}_{2}^{\Cal{L}^{+}}\rangle$,
and so $\Cal{N}_{1}^{+}\Cal{N}_{2}^{+}\leq(\Cal{N}_{1}\Cal{N}_{2})^{+}$. The
reverse inclusion follows from the observation that
$\Cal{N}_{1}\Cal{N}_{2}\leq\Cal{L}\cap\Cal{N}_{1}^{+}\Cal{N}_{2}^{+}$.

Set $R=\Cal{N}_{1}\cap\Cal{N}_{2}$ and suppose $R\leq S$. We have
$R\trianglelefteq\Cal{L}$, so 6.2 yields $R\trianglelefteq\Cal{L}^{+}$.
Observing that

$(\Cal{N}_{1}^{+}\cap\Cal{N}_{2}^{+})\cap\Cal{L}=(\Cal{N}_{1}^{+}\cap\Cal{L})\cap(\Cal{N}_{2}^{+}\cap\Cal{L})=\Cal{N}_{1}\cap\Cal{N}_{2}$

and that $R\cap\Cal{L}=\Cal{N}_{1}\cap\Cal{N}_{2}$, we conclude from 7.3 that
$R=\Cal{N}_{1}^{+}\cap\Cal{N}_{2}^{+}$. ∎

We wish also to have a version of Theorem A for localities which are
homomorphic images of proper localities.

###### Definition Hypothesis 7.5

$(\Cal{L},\Delta,S)$ is a locality on $\Cal{F}$, and
$\Cal{N}\trianglelefteq\Cal{L}$ is a partial normal subgroup of $\Cal{L}$. Set
$T=S\cap\Cal{N}$. Let $(\overline{\Cal{L}},\overline{\Delta},\overline{S})$ be
the quotient locality $\Cal{L}/\Cal{N}$, and let
$\rho:\Cal{L}\to\overline{\Cal{L}}$ be the quotient map. Set
$\overline{\Cal{F}}:=\Cal{F}_{\overline{S}}(\overline{\Cal{L}})$.

The following lemma clarifies the relationship between the fusion system
$\Cal{F}$ of $\Cal{L}$ and the fusion system of $\overline{\Cal{L}}$.

###### Lemma 7.6

Assume the setup of 7.5.

###### Demonstration Proof

As $\rho$ is a projection we have $\overline{\Delta}=\\{P\rho\mid
P\in\Delta\\}$ by definition I.5.4. Thus $\sigma:S\to\overline{S}$ is a
surjective homomorphism of groups. For each $w\in\mathbb{W}(\Cal{L})$ and each
subgroup $X$ of $S_{w}$, write $X^{w}$ for the image of $U$ under the
composite $c_{w}$ of the conjugation maps given sequentially by the entries of
$w$. Similarly define $\overline{X}^{\overline{w}}$ when
$\overline{w}\in\mathbb{W}(\overline{\Cal{L}})$ and
$\overline{X}\leq\overline{S}_{\overline{w}}$. By definition, $Hom(\Cal{F})$
is the set of all mappings $c_{w}:X\to Y$ with $X^{w}\leq Y\leq S$, and
similarly for $Hom(\overline{\Cal{F}})$. If $w=(g_{1},\cdots,g_{n})$ there is
then a commutative diagram of homomorphisms:

$\begin{CD}X@>{c_{w}}>{}>Y\\\ @V{\sigma}V{}V@V{}V{\sigma}V\\\
\overline{X}@>{}>{c_{w\rho^{*}}}>\overline{Y}\end{CD}$

and thus $\sigma$ is fusion-preserving. Suppose now that $X$ and $Y$ contain
$T$, and let $\overline{w}\in\mathbb{W}(\overline{\Cal{L}})$ with
$\overline{X}^{\overline{w}}\leq\overline{Y}$. By I.3.11 there exists a
$\rho^{*}$-preimage $w$ of $\overline{w}$ with
$w\in\mathbb{W}(N_{\Cal{L}}(T))$. Then $X^{w}\leq Y$, and $\sigma\mid_{X}\circ
c_{\overline{w}}=c_{w}\circ\sigma\mid Y$ as maps from $X$ to $\overline{Y}$.
That is, the $\overline{\Cal{F}}$-homomorphism
$c_{\overline{w}}:\overline{X}\to\overline{Y}$ is in the image of
$\sigma_{X,Y}$, and thus (a) holds.

Point (b) is given by the observation that if $X$ is a subgroup of $S$
containing $T$ then $\overline{N_{S}(X)}=N_{\overline{S}}(\overline{X})$. Now
let $\overline{X}\in\overline{\Cal{F}}^{c}$, let $X$ be the preimage of
$\overline{X}$ in $S$, and let $Y$ be an $\Cal{F}$-conjugate of $X$. Then
$\overline{Y}$ is an $\overline{\Cal{F}}$-conjugate of $\overline{X}$, so
$C_{\overline{S}}(\overline{Y})\leq\overline{Y}$, and hence $C_{S}(Y)\leq Y$.
This proves (c). Now assume further that
$\overline{X}\in\overline{\Cal{F}}^{cr}$, and let $Y\in X^{\Cal{F}}$ with $Y$
fully normalized in $\Cal{F}$. Then $\overline{Y}$ is fully normalized in
$\overline{\Cal{F}}$ by (b). Set $Q=O_{p}(N_{\Cal{F}}(Y))$. Then
$\overline{Q}\leq O_{p}(N_{\overline{\Cal{F}}}(\overline{Y}))$ by (a), so
$\overline{Q}=\overline{Y}$, and hence $Q=Y$. This shows that
$X\in\Cal{F}^{cr}$ and proves (d). Point (e) is immediate from (d). ∎

If $(\Cal{L},\Delta,S)$ is a proper locality and
$\Cal{N}\trianglelefteq\Cal{L}$ then the quotient locality $\Cal{L}/\Cal{N}$
need not be proper, just as a homomorphic image of a group of characteristic
$p$ need not be of characteristic $p$. Thus Theorem A not directly applicable
to homomorphic images of proper localities.

###### Theorem 7.7

Let $(\Cal{L},\Delta,S)$ be a proper locality on $\Cal{F}$, let
$\Cal{N}\trianglelefteq\Cal{L}$, let $\overline{\Cal{L}}=\Cal{L}/\Cal{N}$ be
the quotient locality, and let $\rho:\Cal{L}\to\overline{\Cal{L}}$ be the
canonical projection. Let $\Delta^{+}$ be an $\Cal{F}$-closed set of subgroups
of $S$ such that $\Delta\subseteq\Delta^{+}\subseteq\Cal{F}^{s}$, and let
$(\Cal{L}^{+},\Delta^{+},S)$ be the expansion of $\Cal{L}$ to $\Delta^{+}$.
Set $\overline{\Delta}^{+}=\\{P\rho\mid P\in\Delta^{+}\\}$, and set
$\overline{\Cal{F}}=\Cal{F}_{\overline{S}}(\overline{\Cal{L}})$.

###### Demonstration Proof

Note first of all that since $\Cal{F}^{cr}\subseteq\Delta$ we have
$\overline{\Cal{F}}^{cr}\subseteq\overline{\Delta}$ by 7.6(e). Notice also
that if $\Delta=\Delta^{+}$ then there is nothing to prove, so we may assume
that $\Delta$ is properly contained in $\Delta^{+}$.

Among all $\Cal{F}$-closed sets $\Delta_{1}$ with
$\Delta\subseteq\Delta_{1}\subseteq\Delta^{+}$, let $\Delta_{1}$ be maximal
subject to the condition that (a) and (b) hold with $\Delta_{1}$ in the role
of $\Delta^{+}$. Then $\Delta_{1}$ is a proper subset of $\Delta^{+}$, and by
replacing $\Delta$ with $\Delta_{1}$ we reduce (as in the proof of Theorem A1)
to the case where $\Delta^{+}=\Delta\cup R^{\Cal{F}}$ for some $R\leq S$. Thus
$\Cal{L}^{+}$ is an elementary expansion of $\Cal{L}$.

Take $R$ to be fully normalized in $\Cal{F}$, and suppose that $T\nleq R$.
Then $RT\in\Delta$, and then $\Delta^{+}\rho=\Delta\rho=\overline{\Delta}$.
Then $(\overline{\Cal{L}},\overline{\Delta},\overline{S})$ is the unique
locality on $\overline{\Cal{F}}$ whose restriction to $\overline{\Delta}$
(namely, itself) is $\overline{\Cal{L}}$, and thus (a) holds in this case. In
order to verify (b) in this case we appeal to Theorem 4.3(c). Thus
$\Cal{L}^{+}$ is the “free amalgamated product” in the category of partial
groups of the “amalgam” given by the inclusion maps $\Cal{L}_{0}\to\Cal{L}$
and $\Cal{L}_{0}\to\Cal{L}_{0}^{+}$. Point (b) will follow if:

Versions of the same three points will need to be verified in the case where
$T\leq R$, and our approach will be to merely sketch the proofs in each case,
leaving some of the entirely mechanical details to the reader. Thus, let
$\Phi$ be the set of triples $(x^{-1},g,y)$ defined following 4.5, let
$\approx$ be the relation on $\Phi$ defined in 4.11, and let
$\rho^{*}:\mathbb{W}(\Cal{L})\to\mathbb{W}(\overline{\Cal{L}})$ be the
$\rho$-induced homomorphism of free monoids. Then $\rho^{*}$ maps $\Phi$ into
$\mathbb{D}(\overline{\Cal{L}})$, and it may be routinely verified that the
composition

$\Phi@>\rho^{*}>>\mathbb{D}(\overline{\Cal{L}})@>\overline{\Pi}>>\overline{\Cal{L}}$

is constant on $\approx$-classes. Thus, there is a mapping
$\rho_{0}^{+}:\Cal{L}_{0}^{+}\to\overline{\Cal{L}}$ given by
$[x^{-1},g,y]\mapsto\overline{\Pi}(\overline{x}^{-1},\overline{g},\overline{y})$.
The verification that $\rho_{0}^{+}$ is a homomorphism is also routine, and
yields (1). Since $\rho^{*}$ maps $\mathbb{D}(\Cal{L})$ onto
$\mathbb{D}(\overline{\Cal{L}})$, (2) is then immediate. As
$Ker(\rho^{+})\cap\Cal{L}=\Cal{N}$, (3) is immediate from Theorem A2. Thus,
the theorem holds if $T\nleq R$.

Assume henceforth that $T\leq R$. Then
$N_{\overline{\Cal{L}}}(\overline{R})=\overline{N_{\Cal{L}}(R))}$, since each
element of $\overline{\Cal{L}}$ has a $\rho$-preimage in $N_{\Cal{L}}(T)$ by
I.3.11. Then $N_{\overline{\Cal{L}}}(\overline{R})$ is a subgroup of
$\overline{\Cal{L}}$ by I.1.15. Thus Hypothesis 4.2 is satisfied, with
$\overline{\Cal{L}}$ and $\overline{R}$ in the roles of $\Cal{L}$ and $R$.
Theorem 4.3 then yields (a). For the same reasons as in the preceding case
(and because the verification of (3) did not in fact make use of the
hypothesis that $T\nleq R$) it now suffices to verify the following two
points.

Let $\Phi$ and $\rho^{*}$ be as above, and define $\overline{\Phi}$ to be the
set of triples $\overline{\phi}=(\overline{x}^{-1},\overline{g},\overline{y})$
such that one has

$\overline{U}@>\overline{x}^{-1}>>\overline{R}@>\overline{g}>>\overline{R}@>\overline{y}>>\overline{V}$

(a sequence of conjugation isomorphisms between subgroups of $\overline{S}$,
labeled by the conjugating elements), with
$N_{\overline{S}}(\overline{U})\leq\overline{S}_{\overline{x}^{-1}}$, and with
$N_{\overline{S}}(\overline{V})\leq\overline{S}_{\overline{y}^{-1}}$. Then
$\rho^{*}$ maps $\Phi$ onto $\overline{\Phi}$ by 5.4(a). It need cause no
confusion to denote also by $\sim$ and $\approx$ the two equivalence relations
on $\overline{\Phi}$ given by direct analogy with 4.6 and 4.11.

Again by means of 7.6(a) one verifies that the restriction of $\rho^{*}$ to
$\Phi$ preserves these equivalence relations, and hence induces a surjective
mapping $\rho_{0}^{+}\Cal{L}_{0}^{+}\to\overline{\Cal{L}}_{0}^{+}$. Here
$\Cal{L}_{0}$ is by definition the set of elements $f\in\Cal{L}$ such that
$S_{f}$ contains an $\Cal{F}$-conjugate of $R$, and any such $f$ is identified
with its $\approx$-class $[f]$, consisting of all those
$\phi\in\Phi\cap\mathbb{D}(\Cal{L})$ such that $\Pi(\phi)=f$. The analogous
definition of $\overline{\Cal{L}}_{0}$ leads to the conclusion that the
restriction of $\rho_{0}^{+}$ to $\Cal{L}_{0}$ is the (surjective) homorphism
$\rho_{0}:\Cal{L}_{0}\to\overline{\Cal{L}}_{0}$ induced by $\rho$. The
verification that $\rho_{0}^{+}$ is a homomorphism, and hence that (4) holds,
is then straightforward.

Set $\mathbb{D}_{0}^{+}=\mathbb{D}(\Cal{L}_{0}^{+})$ and
$\overline{\mathbb{D}}_{0}^{+}=\mathbb{D}(\overline{\Cal{L}_{0}}^{+})$, let
$\overline{w}\in\overline{\mathbb{D}}_{0}^{+}$, and write
$\overline{w}=([\overline{\phi}_{1}],\cdots,[\overline{\phi}_{n}])$. Here
$[\overline{\phi}_{i}]$ is the $\approx$-class of an element
$\overline{\phi}_{i}=(\overline{x}_{i}^{-1},\overline{g}_{i},\overline{y}_{i})$
in $\overline{\Phi}$, and the representatives $\overline{\phi}_{i}$ may be
chosen so that the sequence
$\overline{\gamma}=(\overline{\phi}_{1},\cdots,\overline{\phi}_{n})$ is a
$\overline{\Gamma}$-form of $\overline{w}$, as defined in 3.11. That is, the
word

$\overline{w}_{\overline{g}}=\overline{\phi}_{1}\circ\cdots\circ\overline{\phi}_{n}$

has the property that $\overline{S}_{\overline{w}_{\overline{\gamma}}}$
contains a $\overline{\Cal{F}}$-conjugate of $\overline{R}$. Let $\phi_{i}$ be
a $\rho^{*}$-preimage of $\overline{\phi}_{i}$ in $\Phi$, set
$\gamma=(\phi_{1},\cdots,\phi_{n})$, let $[\phi_{i}]$ be the $\approx$-class
of $\phi_{i}$ (in $\Cal{L}_{0}^{+}$), and set
$w=([\phi_{1}],\cdots,[\phi_{n}])$. One verifies that $\gamma$ is a
$\Gamma$-form of $w$, and hence that $w\in\mathbb{D}_{0}^{+}$. Since
$\rho^{*}$ maps $w_{\gamma}$ to $\overline{w}_{\overline{\gamma}}$, it follows
that $\rho_{0}^{+}$ induces a surjection as required in (5). Thus (5) holds,
and the proof is complete. ∎

The preceding result is essentially a generalization of Theorem A1 to
homomorphic images of proper localities. Here is the corresponding version of
Theorem A2.

###### Theorem 7.8

Let the hypothesis and the setup be as in the preceding theorem, let
$\mathfrak{N}$ be the set of partial normal subgroups of $\Cal{L}$ containing
$Ker(\rho)$, and let $\mathfrak{N}^{+}$ be the set of partial normal subgroups
of $\Cal{L}^{+}$ containing $Ker(\rho^{+})$. Also, let
$\overline{\mathfrak{N}}$ be the set of partial normal subgroups of
$\overline{\Cal{L}}$, and let $\overline{\mathfrak{N}}^{+}$ be the set of
partial normal subgroups of $\overline{\Cal{L}}^{+}$. Then there is a
commutative diagram of inclusion-preserving bijections:

$\begin{CD}\mathfrak{N}@>{\eta}>{}>\mathfrak{N}^{+}\\\
@V{\beta}V{}V@V{}V{\beta^{+}}V\\\
\overline{\mathfrak{N}}@>{}>{\overline{\eta}}>\overline{\mathfrak{N}}^{+}\end{CD}$

where $\eta$ is the map $\Cal{N}\mapsto\Cal{N}^{+}$, $\beta$ is the map
$\Cal{N}\mapsto\Cal{N}\rho$, and $\beta^{+}$ is the map
$\Cal{N}^{+}\mapsto\Cal{N}^{+}\rho^{+}$. Moreover:

###### Demonstration Proof

By I.5.7 there are inclusion-preserving bijections

$\beta:\mathfrak{N}\to\overline{\mathfrak{N}}\ \ \text{and}\ \
\beta^{+}:\mathfrak{N}^{+}\to\overline{\mathfrak{N}}^{+},$

by which a partial normal subgroup is sent to its image in
$\overline{\Cal{L}}$ (or in $\overline{\Cal{L}}^{+}$) under the canonical
projection $\rho$ (or $\rho^{+}$). Theorem A2 yields the inclusion-preserving
bijection $\eta:\mathfrak{N}\to\mathfrak{N}^{+}$ which sends
$\Cal{N}\in\mathfrak{N}$ to the unique $\Cal{N}^{+}\trianglelefteq\Cal{L}^{+}$
such that $\Cal{N}^{+}$ intersects $\Cal{L}$ in $\Cal{N}$. We obtain the above
commutative square by taking $\overline{\eta}$ to be
$\beta^{-1}\circ\eta\circ\beta^{+}$, so it remains to prove (a) and (b),

We have $Ker(\rho)=\Cal{L}\cap Ker(\rho^{+})$ since $\rho$ is the restriction
of $\rho^{+}$ to $\Cal{L}$. Thus $Ker(\rho^{+})$ is a partial normal subgroup
of $\Cal{L}^{+}$ which intersects $\Cal{L}$ in $Ker(\rho)$. As this condition
characterized $Ker(\rho)\eta$, we have (a).

Fix $\Cal{N}\in\mathfrak{N}$, set $\overline{\Cal{N}}=\Cal{N}\rho$, and set
$\overline{\Cal{N}}^{+}=\Cal{N}^{+}\rho^{+}$. Thus
$\overline{\Cal{N}}^{+}=\overline{\Cal{N}}\overline{\eta}$. Identify
$\overline{\Cal{L}}^{+}$ with $\Cal{L}^{+}/Ker(\rho^{+})$, let
$g\in\Cal{N}^{+}$, and let $[g]$ be the maximal coset of $Ker(\rho^{+})$ in
$\Cal{L}^{+}$ containing $g$. Then $[g]=Ker(\rho^{+})h$ for some $h\in[g]$
such that $h$ is $\uparrow$-maximalin $\Cal{L}^{+}$ with respect to
$Ker(\rho^{+})$. The Splitting Lemma (I.4.11) yields $x\in Ker(\rho^{+})$ such
that $(x,h)\in\mathbb{D}(\Cal{L}^{+})$, $xh=g$, and $S_{(x,h)}=S_{g}$. Then
$(x^{-1},x,h)\in\mathbb{D}(\Cal{L}^{+})$, and $\mathbb{D}$-associativity
yields $h=x^{-1}g$. As $Ker(\rho)\leq\Cal{N}$, (a) implies that
$Ker(\rho^{+})\leq\Cal{N}^{+}$. Thus $h\in\Cal{N}^{+}$, and then
$[g]\subseteq\Cal{N}^{+}$. Now suppose that also $[g]\in\overline{\Cal{L}}$.
As $\overline{\Cal{L}}$ is the image of $\Cal{L}$ under $\rho^{+}$, we may
then assume that $g$ was chosen so that $g\in\Cal{L}$. That is, we may take
$g\in\Cal{L}\cap N^{+}$. Then $g\in\Cal{N}$, and this shows that
$\overline{\Cal{N}}^{+}$ intersects $\overline{\Cal{L}}$ in
$\overline{\Cal{N}}$. The uniqueness of $\overline{\Cal{N}}^{+}$ for this
property is given by the commutativity of the displayed diagram, and so (b)
holds. ∎

Section 8: The fusion system of a proper locality

This section will build on the results of section 3, concerning fusion systems
of centric localities, using Theorem A1 to make the transition from proper
centric localities to proper localities in general. As in section 3, many of
the ideas here are taken from [5A]. We have relied also on [He2] for the
devolopment of the “subcentric” locality associated with a proper locality.

First, we need the following fundamental result, which is a version (for
discrete localities) of Theorem 2.2 in [5A].

###### Theorem 8.1

Let $(\Cal{L},\Delta,S)$ be a locality on $\Cal{F}$, and let
$\mathbb{P}(\Cal{F})$ be the set of all $\Cal{F}$-centric subgroups $P$ of $S$
such that for some $\Cal{F}$-conjugate $Q$ of $P$ with $Q$ fully normalized in
$\Cal{F}$ $($with respect to the stratification induced from $\Cal{L})$, we
have:

$Syl_{p}(Aut_{\Cal{F}}(Q))\neq\emptyset\ \ \text{and}\ \
Inn(Q)=O_{p}(Aut_{\Cal{F}}(Q))\cap Aut_{S}(Q).$ $None$

Assume:

Then $\mathbb{P}(\Cal{F})=\Cal{F}^{cr}$, and if $S$ has the normalizer-
increasing property then $\Cal{L}$ is a proper locality.

###### Demonstration Proof

The first step will be to show:

Let $Q\in\mathbb{P}(\Cal{F})$ be given. In order to show that
$Q\in\Cal{F}^{cr}$ we may assume (since both $\Cal{F}^{cr}$ and
$\mathbb{P}(\Cal{F})$ are $\Cal{F}$-invariant) that $Q$ is fully normalized in
$\Cal{F}$ and satisfies (*). Set $M=N_{\Cal{L}}(Q)$ and set $R=N_{S}(Q)$. Here
$Q\in\Delta$ by (1), so $\Cal{F}_{R}(M)=N_{\Cal{F}}(Q)$, and I.3.10 yields
$R\in Syl_{p}(M)$. As $Q$ is $\Cal{F}$-centric, (2) yields $C_{M}(Q)=Z(Q)$,
and thus $Aut_{\Cal{F}}(Q)$ may be identified with $M/Z(Q)$. As $R$ is a Sylow
$p$-subgroup of $M$ we obtain $Aut_{R}(Q)\in Syl_{p}(Aut_{S}(Q)$, and then (*)
yields $Q=O_{p}(M)$. Then $Q=O_{p}(N_{\Cal{F}}(Q))$as $M$ is of characteristic
$p$, and thus $Q\in\Cal{F}^{cr}$. This shows
$\mathbb{P}(\Cal{F})\subseteq\Cal{F}^{cr}\cap\Delta$.

Next, let $Q\in\Cal{F}^{cr}\cap\Delta$ be given. In order to show that
$Q\in\mathbb{P}(\Cal{F})$ we may assume that $Q$ is fully normalized in
$\Cal{F}$. With $M$ and $R$ defined as above, we again have
$Q=O_{p}(M)=O_{p}(N_{\Cal{F}}(R))$, and again $Aut_{\Cal{F}}(Q)$ may be
identified with $M/Z(Q)$. Then $Inn(Q)=O_{p}(Aut_{\Cal{F}}(Q))$, so $Q$
satisfies (*), and the proof of (3) is complete.

If $\mathbb{P}(\Cal{F})=\Cal{F}^{cr}$ then (1), (2), and the condition that
$S$ have the normalizer-increasing property are the same as the conditions
(PL1) through (PL3) in the definition (6.7) of proper locality, and there is
then nothing more to prove. Thus, we may assume that
$\mathbb{P}(\Cal{F})\neq\Cal{F}^{cr}$, and then (3) yields
$\Cal{F}^{cr}\nsubseteq\Delta$. In particular, $\Cal{F}^{c}\nsubseteq\Delta$.

By Zorn’s Lemma there exists an expansion $(\Cal{L}^{+},\Delta^{+},S)$ of
$\Cal{L}$ which is maximal subject to $\Cal{F}_{S}(\Cal{L}^{+})=\Cal{F}$, and
such that all $\Cal{L}^{+}$-normalizers of members of $\Delta^{+}$ are of
characteristic $p$. We may then take $\Cal{L}=\Cal{L}^{+}$. Let
$\widetilde{\Delta}$ be the set of all subgroups $X\leq S$ such that
$X^{*}\in\Delta$. Then $\mathbb{D}(\Cal{L})$ is the set of all
$w\in\mathbb{W}(\Cal{L})$ such that $S_{w}$ contains a member of
$\widetilde{\Delta}$, so we may take $\Delta=\widetilde{\Delta}$. Among all
$Q\in\Cal{F}^{c}$ with $Q\notin\Delta$, take $Q$ so that $dim(Q)$ is as large
as possible. take $Q$ to be fully normalized in $\Cal{F}$. As
$\Delta=\widetilde{\Delta}$ we then have $Q\in\Omega$. Any strict overgroup of
$Q$ in $S$ is then of greater dimension than $Q$, and is therefore in
$\Delta$. We are free to replace $Q$ with any $\Cal{F}$-conjugate of $Q$, so
we may assume that $Q$ is fully normalized in $\Cal{F}$.

Set $R=N_{S}(Q)$, set $\Cal{M}=N_{\Cal{L}}(Q)$, and let $\Sigma$ be the set of
strict overgroups of $Q$ in $R$. For any
$w\in\mathbb{W}(\Cal{M})\cap\mathbb{D}(\Cal{L})$ we have $Q<S_{w}$ as
$S_{w}\in\Delta$, and then $Q<N_{S_{w}}(Q)$ by I.3.2. Thus
$N_{S_{w}}(Q)\in\Sigma$. As $\Cal{M}$ is a partial subgroup of $\Cal{L}$ we
obtain:

$\mathbb{D}(\Cal{M})=\mathbb{W}(\Cal{M})\cap\mathbb{D}(\Cal{L})=\\{w\in\mathbb{W}(\Cal{M})\mid
R\cap S_{w}\in\Sigma\\}.$

Since $\Sigma$ is closed under $\Cal{M}$-conjugation into $R$ and is closed
with respect to overgroups in $R$, we conclude that $(\Cal{M},\Sigma)$ is an
objective partial group. As $Q$ is fully normalized, I.3.9 implies that $R$ is
a maximal $p$-subgroup of $\Cal{M}$, and so $(\Cal{M},\Sigma,R)$ is a pre-
locality. The poset $\Omega_{R}(\Cal{M})$ is finite-dimensional by I.2.15, so
$(\Cal{M},\Sigma,R)$ is a locality.

Let $g\in C_{\Cal{L}}(Q)$ and set $X=S_{g}\cap R$. Then $XX^{g}\subseteq S$,
and so $[X,g]$ is defined and is a $p$-subgroup of $C_{S}(Q)$. As $Q$ is
centric in $\Cal{F}$ we get $[X,g]\leq Z(Q)$, so $X$ is $g$-invariant. As
$X\in\Sigma\subseteq\Delta$, $g$ is an element of the locally finite group
$N_{\Cal{L}}(X)$. of characteristic $p$. Write $g=ab$ with $a$ of order prime
to $p$ and with $b$ a $p$-element. Then $a$ is a power of $g$, so $[X,a]\leq
Z(Q)$ and then $[X,a]=\mathbb{1}$ by 6.3. Set $Y=O_{p}(N_{\Cal{L}}(X))$. Then
$[Y,a]\leq C_{Y}(X)$, and so $[Y,a]\leq Z(X)$ as $X$ is $\Cal{F}$-centric.
Thus $[Y,a]=\mathbb{1}$, and then $a=\mathbb{1}$ since $N_{\Cal{L}}(X)$ is of
characteristic $p$. Thus $Z(Q)\langle g\rangle$ is a $p$-subgroup of
$C_{\Cal{L}}(Q)$. Here $C_{R}(Q)=Z(Q)$, and since
$C_{\Cal{L}}(Q)\trianglelefteq\Cal{M}$ it follows from I.4.1(c) that $Z(Q)$ is
a maximal $p$-subgroup of $C_{\Cal{L}}(Q)$. We conclude that $g\in Z(Q)$, and
so:

We aim next to show:

Since $N_{S}(P)\in\Delta$ for $P\in Q^{\Cal{F}}$, (5) is equivalent to saying
that $Q$ is normalizer-inductive in $\Cal{F}$ (as defined in 2.8). The proof
of (5) will then be given by 3.17, in the form in which 3.17 was formulated
via the conditions (A) through (D), once these conditions have been verified
to hold in the present context. For ease of reference, we repeat those
conditions here.

Of course, (A) is immediate from the prevailing hypotheses. Taking $\Gamma$ to
be $\Delta$ and taking $V=Q$, we then obtain (B) from 3.1, and again (D) is
immediate. It thus remains to verify (C). So, let $X\in\Delta$ be given with
$X$ fully normalized in $\Cal{F}$, and set $R=N_{S}(X)$. Then $\Delta_{X}$ is
the set of all overgroups of $X$ in $R$. In particular we have
$X\in\Delta_{X}$, so $\Cal{L}_{X}$ is the group $N_{\Cal{L}}(X)$, whence
$\Cal{F}_{R}(\Cal{L}_{X})=N_{\Cal{F}}(X)$, and thus
$(\Cal{L}_{X},\Delta_{X},R)$ is a locality on $N_{\Cal{F}}(X)$. Since
$X\in\Cal{F}^{c}$ it follows that for each $w\in\mathbb{W}(\Cal{L}_{X})$ the
group $N_{S_{w}}(X)$ is centric in $\Cal{F}$, hence centric also in
$N_{\Cal{F}}(X)$. Thus for $X\in\Delta$ we may view the group $\Cal{L}_{X}$ as
a centric locality on $N_{\Cal{F}}(X)$. Theorem 3.9 then yields (C), and so
(5) holds.

The last main step is:

Suppose for the moment that (6) holds. The mapping $\Cal{M}\to
Aut_{\Cal{F}}(Q)$ given by $x\mapsto c_{x}$ is then a surjective homomorphism,
and the quotient locality $\Cal{M}/C_{\Cal{L}}(Q)$ is therefore a group. That
is, $\Cal{M}/Z(Q)$ is a group, by (5). Since $\mathbb{D}(\Cal{M})$ is the set
of all $w\in\mathbb{W}(\Cal{M})$ such that $w$ projects to a member of
$\mathbb{D}(\Cal{M}/Z(Q)$, we obtain
$\mathbb{D}(\Cal{M})=\mathbb{W}(\Cal{M})$, and thus $\Cal{M}$ is a subgroup of
$\Cal{L}$. Hypothesis 4.2 then holds with $Q$ in the role of $R$. By Theorem
4.3 there is an expansion $(\Cal{L}^{+},\Delta^{+},S)$ of $\Cal{L}$ to a
locality on $\Cal{F}$, with $\Delta^{+}=\Delta\cup Q^{\Cal{F}}$ and with
$N_{\Cal{L}^{+}}(Q)=\Cal{M}$. As this contradicts the maximality of $\Delta$,
it therefore remains only to prove (6).

Set $\Cal{F}_{Q}=\Cal{F}_{R}(\Cal{M})$. Assuming now that (6) is false, there
exists an $N_{\Cal{F}}(Q)$-isomorphism $\phi:A\to B$ such that $\phi$ is not
an $\Cal{F}_{Q}$-isomorphism. As $\phi$ extends to an
$N_{\Cal{F}}(Q)$-isomorphism $AQ\to BQ$ we may assume $Q\leq A$. If $Q<A$ then
$A\in\Delta$, so $\phi$ given by conjugation by an element of $\Cal{M}$, and
$\phi$ is then an $\Cal{F}_{Q}$-homomorphism. Thus $Q=A$ and $\phi\in
Aut_{\Cal{F}}(Q)$.

Let $w\in\mathbb{W}(\Cal{L})$ with $Q\leq S_{w}$ and with $\phi=c_{w}$, and
write $w=(g_{1},\cdots,g_{n})$. Set $Q_{0}=Q$, and for all $i$ with $1\leq
i\leq n$ set $Q_{i}=(Q_{i-1})^{g_{i}}$. Thus $Q_{n}=Q^{w}=Q$. By (5) there
exist elements $a_{i}\in\Cal{L}$ such that $N_{S}(Q_{i})\leq S_{a_{i}}$ and
with $(Q_{i})^{a_{i}}=Q$, and where we may take $a_{0}=a_{n}=\mathbb{1}$. For
each $i$ with $1\leq i\leq n$ set $u_{i}=(a_{i-1}^{-1},g_{i},a_{i})$, and set

$X_{i}=N_{S_{g_{i}}}(Q_{i-1})^{a_{i-1}}.$

Then $Q<X_{i}$, so $X_{i}\in\Delta$, and thus $u_{i}\in\mathbb{D}(\Cal{L})$
via $X_{i}$. Set $h_{i}=\Pi(u_{i})$ and set $w^{\prime}=(h_{1},\cdots,h_{n})$.
Then $w^{\prime}\in\mathbb{W}(N_{\Cal{L}}(Q))$ and $c_{w^{\prime}}$ is equal
to $c_{w}$ as automorphisms of $Q$. This shows that $\phi$ is an
$\Cal{F}_{Q}$-automorphism, so the proof of (6), and of the theorem, is now
complete. ∎

###### Definition Definition 8.2

Let $\Cal{F}$ be a fusion system on the countable, locally finite $p$-group
$S$, and assume that for each subgroup $X\leq S$ the group $Aut_{\Cal{F}}(X)$
is countable and locally finite. Let $Y$ be a subgroup of $S$. For any
$\Cal{F}$-isomorphism $\alpha:X\to Y$ define $N_{\alpha}$ to be the set of all
$g\in N_{S}(X)$ such that the automorphism $\alpha^{-1}\circ c_{g}\circ\alpha$
of $Y$ is in $Aut_{S}(Y)$.

$\Cal{F}$ is saturated if:

###### Theorem 8.3

Let $(\Cal{L},\Delta,S)$ be a proper locality on $\Cal{F}$, let $A\leq S$ be
fully normalized in $\Cal{F}$, and let $B$ be fully centralized in $\Cal{F}$.

###### Demonstration Proof

By Theorem A1, the restriction of $\Cal{L}$ to the $\Cal{F}$-closure of
$\Cal{F}^{cr}$ is a proper locality $\Cal{L}_{0}$ on $\Cal{F}$, as is the
expansion of $\Cal{L}_{0}$ to any locality $(\Cal{L}_{1},\Delta_{1},S)$ with
$\Delta_{1}=\Cal{F}^{c}$. Without loss of generality we may then take
$\Cal{L}=\Cal{L}_{1}$ and $\Delta=\Cal{F}^{c}$. Then Theorem 3.9 yields (a),
and yields also a centric locality $(\Cal{L}_{A},\Delta_{A},N_{S}(A))$ with an
inclusion homomorphism $\Cal{L}_{A}\to N_{\Cal{L}}(A)$, and a centric locality
$(\Cal{C}_{B},\Sigma_{B},C_{S}(B))$ on $C_{\Cal{F}}(B)$ with an inclusion
homomorphism $\Cal{C}_{B}\to C_{\Cal{L}}(B)$. As in 3.7 and 3.8 we may take
$\Delta_{A}=\\{P\in\Delta\mid A\trianglelefteq P\\}$ and $\Sigma_{B}=\\{Q\leq
C_{S}(B)\mid Z(B)\leq Q\ \text{and}\ \ BQ\in\Delta\\}$.

In order to complete the proofs of (d) and (e) it remains to show that
$\Cal{L}_{A}$ and $\Cal{C}_{B}$ are proper. Since $N_{S}(A)$ and $C_{S}(B)$
inherit the normalizer-increasing property from $S$ it then suffices to show
that $N_{\Cal{L}_{A}}(P)$ is of characteristic $p$ for all $P\in\Delta_{A}$
and that $N_{\Cal{C}_{B}}(Q)$ is of characteristic $p$ for all
$Q\in\Sigma_{B}$. So, let $P\in\Delta_{A}$ be given. Then $P\in\Delta$, and
$N_{\Cal{L}_{A}}(P)$ is the group $N_{N_{\Cal{L}}(P)}(A)$, which is of
characteristic $p$ by 6.5(b). Similarly, if $Q\in\Sigma_{B}$ then
$BQ\in\Delta$ then $N_{\Cal{C}_{B}}(Q)$ is the group $C_{N_{\Cal{L}}(BQ)}(B)$.
Then $N_{\Cal{C}_{B}}(Q)\trianglelefteq N_{N_{\Cal{L}}(BQ)}(B)$, and then
6.5(a) shows that $N_{\Cal{C}_{B}}(Q)$ is of characteristic $p$. Thus (d) and
(e) hold.

Next, set $\Cal{K}=C_{\Cal{L}_{A}}(A)$, $P=C_{S}(A)A$, and
$H=N_{\Cal{L}_{A}}(P)$. Then $\Cal{K}\trianglelefteq\Cal{L}_{A}$, and the
Frattini Lemma (I.4.10) yields $\Cal{L}_{A}=H\Cal{K}$. As $A$ is fully
normalized, (a) implies that $A$ is fully centralized, and then
$P\in\Cal{F}^{c}$ by 3.5(c). Thus $P\in\Delta_{A}$, so $H$ is a subgroup of
$\Cal{L}$, and then $H$ is countable and locally finite. Here
$Aut_{\Cal{F}}(A)\cong\Cal{L}_{A}/\Cal{K}\cong H/C_{H}(A)$, so
$Aut_{\Cal{F}}(A)$ is countable and locally finite. Since
$Aut_{\Cal{F}}(A)\cong Aut_{\Cal{F}}(A^{\prime})$ for every
$\Cal{F}$-conjugate $A^{\prime}$ of $A$, we have point (b).

We may view $H$ as a locality $(H,\Delta_{A},N_{S}(A))$ (where finite-
dimensionality is given by I.2.15(a)). Then $N_{S}(A)\in Syl_{p}(H)$ by 3.8,
and so $A$ is fully automized in $\Cal{F}$. To show that $A$ is receptive, let
$A^{\prime}\in A^{\Cal{F}}$ and let $\alpha:A^{\prime}\to A$ be an
$\Cal{F}$-isomorphism. By (a) there exists an $\Cal{F}$-homomorphism
$\phi:N_{S}(A^{\prime})\to N_{S}(A)$ with $A^{\prime}\phi=A$. Let $\psi$ be
the restriction of $\phi$ to $A^{\prime}$, and set
$\eta=\psi^{-1}\circ\alpha$. Thus $\eta\in Aut_{\Cal{F}}(A)$. Let $x\in
N_{S}(A^{\prime})$. Then $c_{x\phi}\in Aut_{S}(A)$. Set $\beta=\psi\circ
c_{x\phi}\circ\psi^{-1}$. Then $\beta\in Aut(A^{\prime})$, and for any $b\in
A^{\prime}$ we compute:

$b\beta=((x^{-1}\psi)(b\psi)(x\psi))\psi^{-1}=x^{-1}bx.$

Thus $\beta=c_{x}$. Now assume that $x\in N_{\alpha}$, and compute

$\eta^{-1}\circ c_{x\phi}\circ\eta=\alpha^{-1}\circ(\psi\circ
c_{x\phi}\circ\psi^{-1})\circ\alpha=\alpha^{-1}\circ c_{x}\circ\alpha\in
Aut_{S}(A),$

which shows that $N_{\alpha}\phi\leq N_{\eta}$. If $\eta$ extends to an
$\Cal{F}$-homomorphism $\overline{\eta}:N_{\eta}\to N_{S}(A)$ then
$\phi\mid_{N_{\alpha}}\circ\overline{\eta}$ is an extension of $\alpha$ to an
$\Cal{F}$-homomorphism $N_{\alpha}\to N_{S}(A)$. Thus, it now suffices to show
that $A$ is receptive in $N_{\Cal{F}}(A)$, in order to show that $A$ is
receptive in $\Cal{F}$. Thus, in proving receptivity for $A$ in $\Cal{F}$ we
may assume that $A\trianglelefteq\Cal{F}$, $A\trianglelefteq\Cal{L}$, and
$\alpha$ is an automorphism of $A$.

Recall that $Aut_{\Cal{F}}(A)=Aut_{H}(A)$, where $H=N_{\Cal{L}}(C_{S}(A)A)$.
We may therefore assume that $\Cal{L}=H$, a group of characteristic $p$. Here
$\alpha=c_{h}$ for some $h\in H$, and for $x\in N_{\alpha}$ we have
$h^{-1}c_{x}h=c_{y}$ for some $y\in S$. Thus $c_{x^{h}}=c_{y}$, so
$x^{h}y^{-1}\in C_{H}(A)$. Thus $(N_{\alpha})^{h}\leq C_{H}(A)S$. Here $S\in
Syl_{p}(C_{H}(A)S)$ by I.3.13(b), so there exists $k\in C_{H}(A)$ with
$(N_{\alpha})^{hk}\leq S$. Then $c_{hk}:N_{\alpha}\to S$ is an extension of
$\alpha$, and thus $A$ is receptive.

The requirement (4) in 8.2, that $\Cal{F}$ be stratified, is given by the
stratification induced from $\Cal{L}$. The proof is thereby complete. ∎

###### Lemma 8.4

Let $(\Cal{L},\Delta,S)$ be a proper locality on $\Cal{F}$ let
$\Cal{N}\trianglelefteq\Cal{L}$ be a partial normal subgroup, set
$T=S\cap\Cal{N}$, and let $\Cal{E}$ be the fusion system
$\Cal{F}_{T}(\Cal{N})$. Then $P\cap T\in\Cal{E}^{c}$ for each
$P\in\Cal{F}^{cr}$.

###### Demonstration Proof

Fix $P\in\Cal{F}^{cr}$, set $U=P\cap T$, and let $V\in U^{\Cal{F}}$ be fully
normalized in $\Cal{F}$. As $\Cal{F}$ is inductive, and since $P\leq
N_{S}(U)$, there is an $\Cal{F}$-conjugate $Q$ of $P$ with $Q\cap T=V$. Set
$H=N_{\Cal{L}}(Q)$. Then $H$ is a subgroup of $\Cal{L}$ of characteristic $p$
(as $\Cal{F}^{cr}\subseteq\Delta$), and $Q=O_{p}(H)$ by 6.9(a).

Set $D=N_{C_{T}(V)}(Q)$. Then $[Q,D]\leq Q\cap T=V$, and so
$[Q,D,D]=\mathbb{1}$. As $Q=O_{p}(H)$ we then have $D\leq Q$ by 6.6(c), and so
$D\leq V$. As $C_{T}(V)$ is $Q$-invariant, and since $C_{T}(V)Q$ inherits the
normalizer-increasing property from $S$, we conclude that $C_{T}(V)\leq V$.
For any $\Cal{F}$-conjugate $X$ of $V$ there exists an $\Cal{F}$-homomorphism
$N_{S}(X)\to N_{S}(V)$ which sends $X\to V$, and hence $C_{T}(X)\leq X$. Thus
$V$ is centric in $\Cal{E}$, as is every $\Cal{F}$-conjugate of $V$; and so
$P\cap T\in\Cal{E}^{c}$ as desired. ∎

The next few results concern the set $\Cal{F}^{s}$ of $\Cal{F}$-subcentric
subgroups of the fusion system $\Cal{F}$ of a proper locality. The proofs will
be variations of proofs of corresponding results in [He2], where $S$ is
finite.

###### Lemma 8.5

Let $\Cal{F}$ be the fusion system on $S$ of a proper locality $\Cal{L}$, and
let $V\leq S$ be fully centralized in $\Cal{F}$. Then $V\in\Cal{F}^{s}$ if and
only if $O_{p}(C_{\Cal{F}}(V))$ is centric in $C_{\Cal{F}}(V)$.

###### Demonstration Proof

Let $U\in V^{\Cal{F}}$ be fully normalized in $\Cal{F}$. As $\Cal{F}$ is
inductive $U$ is then fully centralized in $\Cal{F}$, and 1.6 yields an
isomorphism $C_{\Cal{F}}(U)\cong C_{\Cal{F}}(V)$ of fusion systems. We may
therefore assume to begin with that $V$ is fully normalized in $\Cal{F}$. Set
$\Cal{F}_{V}=N_{\Cal{F}}(V)$ and $R=O_{p}(\Cal{F}_{V})$. Also, set
$\Cal{C}_{V}=C_{\Cal{F}}(V)$ and $Q=O_{p}(\Cal{C}_{V})$.

Suppose first that the lemma holds with $\Cal{F}_{V}$ in place of $\Cal{F}$.
That is, suppose that $R$ is centric in $\Cal{F}_{V}$ if and only if $Q$ is
centric in $\Cal{C}_{V}$. Since $R\in(\Cal{F}_{V})^{c}$ if and only if
$R\in\Cal{F}^{c}$ by 3.6(b), we then have the lemma in general. Thus, we are
reduced to the case where $V\trianglelefteq\Cal{F}$, and $R=O_{p}(\Cal{F})$.

We are given a proper locality $(\Cal{L},\Delta,S)$ on $\Cal{F}$, and by
Theorem A1 we may assume that $\Cal{F}^{c}\subseteq\Delta$. Suppose that
$V\in\Cal{F}^{s}$. Then $R\in\Cal{F}^{c}$, so $R\in\Delta$, and $\Cal{L}$ is
the group $N_{\Cal{L}}(R)$. Notice that $[R,C_{S}(VQ)]\leq C_{R}(VQ)\leq Q$,
and that both $R$ and $Q$ are normal subgroups of $\Cal{L}$. Then
$C_{S}(VQ)\leq R$ by 6.5(c). Then also

$C_{C_{S}(V)}(Q)=C_{S}(VQ)\leq C_{R}(V)\leq Q,$

and thus $Q$ is centric in $C_{\Cal{F}}(V)$. On the other hand, assuming that
$Q$ is centric in $C_{\Cal{F}}(V)$, we obtain

$C_{S}(R)\leq C_{S}(VQ)\leq Q\leq R,$

and so $R\in\Cal{F}^{c}$, as required. ∎

###### Corollary 8.6

Let $\Cal{F}$ be the fusion system of a proper locality. Then

$\Cal{F}^{cr}\subseteq\Cal{F}^{c}\subseteq\Cal{F}^{q}\subseteq\Cal{F}^{s}.$

###### Demonstration Proof

We have $\Cal{F}^{cr}\subseteq\Cal{F}^{c}$ by definition. Let
$(\Cal{L},\Delta,S)$ be a proper locality on $\Cal{F}$, and $P\leq S$ be fully
normalized in $\Cal{F}$. If $P\in\Cal{F}^{c}$ then $C_{\Cal{F}}(P)$ is the
trivial fusion system on $Z(P)$, and so $P\in\Cal{F}^{q}$. Now suppose instead
that we are given $P\in\Cal{F}^{q}$. Then $O_{p}(C_{\Cal{F}}(Q))=C_{S}(Q)$,
and then $P\in\Cal{F}^{s}$ by 8.5. ∎

###### Corollary 8.7

Let $\Cal{F}$ be the fusion system on $S$ of a proper locality, and let $V\leq
S$ with $V$ fully normalized in $\Cal{F}$. Then

$\\{Q\in N_{\Cal{F}}(V)^{s}\mid V\leq Q\\}\subseteq\Cal{F}^{s}.$

###### Demonstration Proof

One need only observe that $C_{\Cal{F}}(V)=C_{C_{\Cal{F}}(V)}(V)$, in order to
obtain the desired result from 8.5. ∎

###### Theorem 8.8

Let $(\Cal{L},\Delta,S)$ be a proper locality on $\Cal{F}$.

###### Demonstration Proof

The points (a) and (b) concern only $\Cal{F}$ and not $\Cal{L}$, so by Theorem
A1 we may take $\Delta=\Cal{F}^{c}$.

We first prove (b). Let $X\leq S$ be given with $X^{*}\in\Cal{F}^{s}$. Since
$\Cal{F}^{s}$ is $\Cal{F}$-invariant by definition we are free to replace
$X^{*}$ by any $\Cal{F}$-conjugate of $X^{*}$, and thus we may assume to begin
with that $X^{*}$ is fully normalized in $\Cal{F}$ and that
$O_{p}(N_{\Cal{F}}(X^{*}))$ is $\Cal{F}$-centric. By 2.4(c) we have
$X^{*}=Y^{*}$ for any $N_{\Cal{F}}(X^{*})$-conjugate $Y$ of $X$, so we are
free to replace $X$ by any $N_{\Cal{F}}(X^{*})$-conjugate of $X$. As $\Cal{F}$
is inductive, 2.11 shows that we may then take $X$ to be fully normalized in
$\Cal{F}$. Set $\Cal{E}=N_{\Cal{F}}(X)$ and set
$\Cal{E}^{*}=N_{\Cal{F}}(X^{*})$. Then $\Cal{E}=N_{\Cal{E}^{*}}(\Cal{E})$. In
order to show that $X$ is subcentric in $\Cal{F}$ it suffices, by 3.6(b), to
show that $O_{p}(\Cal{E})$ is centric in $\Cal{E}$, so we may assume that
$\Cal{L}$ is the locality $\Cal{L}_{X^{*}}$ given by Theorem 3.9(b). That is,
we may assume $X^{*}\trianglelefteq\Cal{L}$. Then
$X^{*}=O_{p}(\Cal{L})\in\Cal{F}^{c}$, and thus $\Cal{L}$ is a group $G$ of
characteristic $p$. Then also $N_{G}(X)$ is of characteristic $p$, by 6.5(b).
As $\Cal{E}=\Cal{F}_{N_{S}(X)}(N_{G}(X))$ we then have
$O_{p}(\Cal{E})\in\Cal{E}^{c}$, as desired. We thus have (b).

In proving (a), note that by definition $\Cal{F}^{s}$ is $\Cal{F}$-invariant
and that $S\in\Cal{F}^{s}$. Thus it only remains to show that $\Cal{F}^{s}$ is
closed with respect to overgroups in $S$. Among all $V\in\Cal{F}^{s}$ such
that some overgroup $Q$ of $V$ in $S$ is not subcentric in $\Cal{F}$, choose
$V$ so that $V$ is fully normalized in $\Cal{F}$, and so that $dim(V)$ is as
large as possible.

Suppose first that $V^{*}\notin\Cal{F}^{s}$. Thus, we may take $Q=V^{*}$. Then
$Q$ is not subcentric in $N_{\Cal{F}}(V)$, by 8.7. But $V$ is subcentric in
$N_{\Cal{F}}(V)$ by 8.6. This shows that the locality $\Cal{L}_{V}$ given by
8.3(d) is a counterexample to (a), and so we may assume
$V\trianglelefteq\Cal{L}$. Then
$V^{*}=O_{p}(\Cal{L})=O_{p}(\Cal{F})=O_{p}(N_{\Cal{F}}(V))$, so
$V^{*}\in\Cal{F}^{c}$, contrary to the hypothesis that
$V^{*}\notin\Cal{F}^{s}$. Thus $V^{*}\in\Cal{F}^{s}$ after all. Here
$Q^{\star}\notin\Cal{F}^{s}$ by (b), we may take $V=V^{\star}$ and
$Q=Q^{\star}$.

As $V<Q$ and $V\Omega$ we have $dim(N_{Q}(V))>dim(V)$, and the maximality of
$dim(V)$ in our choice of $V$ then implies that $N_{Q}(V)\notin\Cal{F}^{s}$.
Thus we may now replace $Q$ with $N_{Q}(V)$. That is, we may assume
$V\trianglelefteq Q$. We are then free to replace $Q$ with any
$N_{\Cal{F}}(V)$-conjugate of $Q$, and so we may assume that $Q$ is fully
normalized in $N_{\Cal{F}}(V)$.

Suppose that $Q\in N_{\Cal{F}}(V)^{s}$. As
$C_{\Cal{F}}(Q)=C_{N_{\Cal{F}}(V)}(Q)$ we then obtain $Q\in\Cal{F}^{s}$ from
8.6. Thus $Q$ is not subcentric in $N_{\Cal{F}}(V)$, and we again reduce to
the situation where $V\trianglelefteq\Cal{L}$ and where $Q$ is fully
normalized in $\Cal{F}$. Here $O_{p}(\Cal{F})=O_{p}(\Cal{F})\in\Cal{F}^{c}$
and hence $\Cal{L}$ is a group of characteristic $p$. Then $N_{\Cal{L}}(Q)$ is
of characteristic $p$ by 6.5(b), and since $Q$ is fully normalized in
$\Cal{F}$ we have also $N_{\Cal{F}}(Q)=\Cal{F}_{N_{S}(Q)}(N_{\Cal{L}}(Q))$.
Then $O_{p}(N_{\Cal{F}}(Q))\in N_{\Cal{F}}(Q)^{c}$, and hence
$O_{p}(N_{\Cal{F}}(Q))\in\Cal{F}^{c}$ by 3.6(b). Thus $Q\in\Cal{F}^{s}$, and
we have (a). ∎

The following result is immediate from 8.8 and Theorem A1.

###### Corollary 8.9

Let $(\Cal{L},\Delta,S)$ be a proper locality on $\Cal{F}$. Then there exists
an expansion of $\Cal{L}$ to a proper locality $(\Cal{L}^{s},\Cal{F}^{s},S)$
on $\Cal{F}$. Moreover, up to a unique isomorphism which restricts to the
identity map on $S$, every expansion of $\Cal{L}$ to a proper locality on
$\Cal{F}$ is a restriction of $\Cal{L}^{s}$. ∎

###### Definition Definition 8.10

If $(\Cal{L},\Delta,S)$ be a proper locality on $\Cal{F}$ then the locality
$(\Cal{L}^{s},\Cal{F}^{s},S)$ is the subcentric closure of $\Cal{L}$.

###### Lemma 8.11

Let $\Cal{F}$ be the fusion system of a proper locality $(\Cal{L},\Delta,S)$,
and let $P\leq S$ be a subgroup of $S$ such that
$O_{p}(\Cal{F})P\in\Cal{F}^{s}$. Then $P\in\Cal{F}^{s}$.

###### Demonstration Proof

Among all counterexamples, choose $P$ so that $dim(P)$ is as large as
possible. Set $Q=O_{p}(\Cal{F})$, and let $\phi:P\to P^{\prime}$ be an
$\Cal{F}$-isomorphism such that $P^{\prime}$ is fully normalized in $\Cal{F}$.
Then $\phi$ extends to an $\Cal{F}$-isomorphism $PQ\to P^{\prime}Q$, so
$P^{\prime}Q\in\Cal{F}^{s}$. Thus we may assume to begin with that $P$ is
fully normalized in $\Cal{F}$. Note that $P^{*}\notin\Cal{F}^{s}$ by 8.8(b),
while $P^{*}Q\in\Cal{F}^{s}$ by 8.8(a). Thus $P^{*}$ is a counterexample to
the lemma, and so we may take $P\in\Omega$.

Set $\Cal{E}=N_{\Cal{F}}(P)$ and set $D=N_{Q}(P)$. Then $D\leq
O_{p}(\Cal{E})$. If $D\leq P$ then $P=PR$, and $P$ is not a counterexample.
Thus $P<DP$, and since $P\in\Omega$ we then have $DP\in\Cal{F}^{s}$ by the
maximality of $dim(P)$ in our choice of $P$. Since $N_{S}(P)\leq N_{S}(PD)$ we
may assume that $P$ was chosen so that also $PD$ is fully normalized in
$\Cal{F}$. Set $R=N_{S}(PD)$. Also, by 8.10 we may assume that
$\Delta=\Cal{F}^{s}$, so that $PD\in\Delta$. Then $G:=N_{\Cal{L}}(PD)$ is a
group of characteristic $p$, $R\in Syl_{p}(G)$, and
$\Cal{F}_{R}(G)=N_{\Cal{F}}(PD)$. Since each $\Cal{E}$-homomorphism extends to
an $N_{\Cal{F}}(PD)$-homomorphism, $\Cal{E}$ is then the fusion system of
$N_{G}(P)$ at $N_{S}(P)$. As $G$ is of characteristic $p$, so is $N_{G}(P)$ by
6.5(b). Thus $O_{p}(\Cal{E})$ is centric in $\Cal{E}$, and hence centric in
$\Cal{F}$ by 3.6(b). Thus $P\in\Cal{F}^{s}$. ∎

We end this section by showing that our notion of radical centric subgroup is
equivalent to a more familiar notion from fusion systems over finite
$p$-groups, in the case of the fusion system of a proper locality.

###### Lemma 8.12

Let $(\Cal{L},\Delta,S)$ be a proper locality on $\Cal{F}$ and let $P\leq S$
be a subgroup of $S$. Assume that $P$ is centric in $\Cal{F}$. Then the
following are equivalent.

###### Demonstration Proof

By Theorem A we may assume without loss of generality that
$\Cal{F}^{c}\subseteq\Delta$. Thus $P\in\Delta$, and then
$C_{\Cal{L}}(P)=Z(P)$ as $\Cal{L}$ is proper. Thus $Aut_{\Cal{F}}(P)$ may be
identified with $N_{\Cal{L}}(P)/Z(P)$, and then
$O_{p}(Aut_{\Cal{F}}(P))=O_{p}(N_{\Cal{L}}(P))/Z(P)$. For any $Q\in
P^{\Cal{F}}$ there exists $g\in\Cal{L}$ with $Q=P^{g}$. Then conjugation by
$g$ is an isomorphism $N_{\Cal{L}}(P)\to N_{\Cal{L}}(Q)$, and we obtain
thereby the equivalence of (1) and (2). ∎

Section 9: $O^{p}_{\Cal{L}}(\Cal{N})$ and $O^{p^{\prime}}_{\Cal{L}}(\Cal{N})$

Throughout this section $(\Cal{L},\Delta,S)$ be a locality,
$\Cal{N}\trianglelefteq\Cal{L}$ is a partial normal subgroup, and we set
$T=S\cap\Cal{N}$.

###### Definition Definition 9.1

Set

$\mathbb{K}=\\{\Cal{K}\trianglelefteq\Cal{L}\mid\Cal{K}T=\Cal{N}\\}\ \
\text{and}\ \
\mathbb{K}^{\prime}=\\{\Cal{K}^{\prime}\trianglelefteq\Cal{L}\mid
T\leq\Cal{K}\\}.$

Then set

$O^{p}_{\Cal{L}}(\Cal{N})=\bigcap\mathbb{K}\ \ \text{and}\ \
O^{p^{\prime}}_{\Cal{L}}(\Cal{N})=\bigcap\mathbb{K}^{\prime}.$

Write $O^{p}(\Cal{L})$ for $O^{p}_{\Cal{L}}(\Cal{L})$, and
$O^{p^{\prime}}(\Cal{L})$ for $O^{p^{\prime}}_{\Cal{L}}(\Cal{L})$.

###### Lemma 9.2

Let $(\Cal{L},\Delta,S)$ be a locality and let $\Cal{N}\trianglelefteq\Cal{L}$
be a partial normal subgroup, such that every subgroup of $\Cal{N}$ is a
$p$-group. Then $\Cal{N}$ is a subgroup of $S$.

###### Demonstration Proof

A partial subgroup of a group is a group, by I.1.8(c), so it suffices to show
that $\Cal{N}$ is contained in $S$. Suppose false, set $T=S\cap\Cal{N}$, and
among all $g\in\Cal{N}$ with $g\notin T$ choose $g$ so that $dim(S_{g}\cap T)$
is as large as possible. Set $P=S_{g}$ and set $X=P\cap T$. Then
$N_{\Cal{N}}(P)$ is a subgroup of $\Cal{N}$, and so $N_{\Cal{N}}(P)$ is a
$p$-group. By I.3.8 there exists $a\in\Cal{L}$ with $N_{\Cal{N}}(P)^{a}\leq
T$, and the Frattini Lemma (I.4.10) shows that we may choose $a\in\Cal{N}$.

Suppose first that $X=T$. Then $g\in N_{\Cal{N}}(P)$ by I.4.1(b). As
$N_{\Cal{N}}(P)\leq N_{\Cal{N}}(T)$ amnd $N_{\Cal{N}}(P)$ is a subgroup of
$\Cal{N}$, so $g$ is a $p$-element of $N_{\Cal{N}}(T)$. But then $g\in T$
since $T$ is a maximal $p$-subgroup of $\Cal{N}$ by I.4.1(c), so in fact
$X<T$. As $P\in\Omega$ we then have $dim(X)<dim(N_{T}(P))$ by I.3.2. Thus
$dim(S_{a}\cap T)>dim(X)$, so $a\in T$, and thus $N_{\Cal{N}}(P)\leq T$. A
similar argument with $g^{-1}$ in place of $g$ shows that
$N_{\Cal{N}}(P^{g})\leq T$, so $X<N_{T}(P)\leq S_{g}\cap T=X$. This
contradiction proves the lemma. ∎

###### Proposition 9.3

Let $(\Cal{L},\Delta,S)$ be a locality, let $\Cal{N}\trianglelefteq\Cal{L}$ be
a partial normal subgroup, and set $T=S\cap\Cal{N}$. Define $\mathbb{K}$ and
$\mathbb{K}^{\prime}$ as in 9.1. Then $O^{p}_{\Cal{L}}(\Cal{N})\in\mathbb{K}$
and $O^{p^{\prime}}_{\Cal{L}}(\Cal{N})\in\mathbb{K}^{\prime}$.

###### Demonstration Proof

It is obvious that $O^{p}_{\Cal{L}}(\Cal{N})$ and
$O^{p^{\prime}}_{\Cal{L}}(\Cal{N})$ are partial normal subgroups of $\Cal{L}$
contained in $\Cal{N}$, and that $T\leq O^{p^{\prime}}_{\Cal{L}}(\Cal{N})$.
Thus $O^{p^{\prime}}_{\Cal{L}}(\Cal{N})\in\mathbb{K}^{\prime}$, and it only
remains to show that $O^{p}_{\Cal{L}}(\Cal{N})T=\Cal{N}$.

Set $\Cal{K}_{0}=O^{p}_{\Cal{L}}(\Cal{N})$, set
$\overline{\Cal{L}}=\Cal{L}/\Cal{K}_{0}$, and adopt the usual “bar”-convention
for images of subsets of $\Cal{L}$ under the canonical projection
$\Cal{L}\to\overline{\Cal{L}}$. Now let $H$ be a subgroup of $\Cal{N}$ and let
$x$ be a $p^{\prime}$-element of $H$. Then $x\in\Cal{K}$ for each
$\Cal{K}\in\mathbb{K}$, and so $x\in\Cal{K}_{0}$. Thus $\overline{H}$ is a
$p$-group. Every subgroup of $\overline{\Cal{N}}$ is of the form
$\overline{H}$ for some subgroup $H$ of $\Cal{N}$ by I.2.16 and I.5.3(c), so
$\overline{\Cal{N}}=\overline{T}$ by 9.2. Thus $\Cal{N}=\Cal{K}_{0}T$, as
required. ∎

###### Lemma 9.4

Let “$*$” be either of the symbols “$p$” or “$p^{\prime}$, let
$(\Cal{L},\Delta,S)$ be a proper locality, and let
$(\Cal{L}^{+},\Delta^{+},S)$ be an expansion of $\Cal{L}$. Let
$\Cal{N}\trianglelefteq\Cal{L}$, and for any partial normal subgroup
$\Cal{K}\trianglelefteq\Cal{L}$ let $\Cal{K}^{+}$ be the corresponding partial
normal subgroup of $\Cal{L}^{+}$ given by Theorem A2. Then
$O^{*}_{\Cal{L}}(\Cal{N})^{+}=O^{*}_{\Cal{L}^{+}}(\Cal{N}^{+})$.

###### Demonstration Proof

Write $\mathbb{K}^{+}$ for the set of all $\Cal{K}^{+}$ with
$\Cal{K}\in\mathbb{K}$. Then

$(\bigcap\mathbb{K})^{+}\leq\bigcap(\mathbb{K}^{+})$

as $\bigcap(\mathbb{K}^{+})$ is a partial normal subgroup of $\Cal{L}^{+}$
containing $\bigcap\mathbb{K}$. The reverse inclusion is given by Theorem A2,
along with the observation that

$\Cal{L}\cap(\bigcap(\mathbb{K}^{+}))=\bigcap\\{\Cal{L}\cap\Cal{K}^{+}\\}_{\Cal{K}\in\mathbb{K}}=\bigcap\mathbb{K}.$

Thus the lemma holds for “$p$”, and the same argument applies to
“$p^{\prime}$”. ∎

###### Lemma 9.5

Let $\Cal{N}$ and $\Cal{M}$ be partial normal subgroups of $\Cal{L}$, with
$\Cal{N}\leq\Cal{M}$. Let “$*$” be either of the symbols “$p$” or
“$p^{\prime}$. Then $O^{*}_{\Cal{L}}(\Cal{N})\leq O^{*}_{\Cal{L}}(\Cal{M})$.

###### Demonstration Proof

Since $O^{p^{\prime}}_{\Cal{L}}(\Cal{M})$ is a partial normal subgroup of
$\Cal{L}$ containg $S\cap\Cal{N}$, we have
$O^{p^{\prime}}_{\Cal{L}}(\Cal{N})\leq O^{p^{\prime}}_{\Cal{L}}(\Cal{M})$ by
definition. Set $\Cal{K}=O^{p}_{\Cal{L}}(\Cal{M})$, set
$\overline{\Cal{L}}=\Cal{L}/\Cal{K}$, and let
$\rho:\Cal{L}\to\overline{\Cal{L}}$ be the canonical projection. Then
$(S\cap\Cal{M})\rho=\Cal{M}\rho\geq\Cal{N}\rho$, and hence
$\Cal{N}\rho=(S\cap\Cal{N})\rho$. Subgroup correspondence (I.4.7) then yields
$\Cal{N}\leq\Cal{K}(S\cap\Cal{N})$, and then
$O^{p}_{\Cal{L}}(\Cal{N})\leq\Cal{K}$ by definition. ∎

Section 10: Henke’s Theorem on extending homomorphisms

In this brief section we adapt [Lemma 3.1 in He3] to our present context. The
hypothesis given below is in some respects weaker than that in [He3], but
Henke’s proof (essentially repeated in 10.3, below) remains valid in every
detail. We have included it for the sake of clarity and completeness.

###### Definition Hypothesis 10.1

We are given localities $(\Cal{L},\Delta,S)$ and $(\Cal{L}^{+},\Delta^{+},S)$
such that $\Delta\subseteq\Delta^{+}$ and such that $\Cal{L}$ is the
restriction of $\Cal{L}^{+}$ to $\Delta$. Set
$\Cal{F}=\Cal{F}_{S}(\Cal{L}^{+})$, and write $(*,\Omega)$ for the
stratification on $\Cal{F}$ induced from $\Cal{L}^{+}$. Assume:

Set $M=N_{\Cal{L}^{+}}(R)$. Assume that there is given a partial group
$\widetilde{\Cal{L}}$, a homomorphism $\alpha:\Cal{L}\to\widetilde{\Cal{L}}$,
and a homomorphism $\gamma_{M}:M\to\widetilde{\Cal{L}}$, such that:

Notice that (2) yields a mapping $\beta:\Cal{L}\cup M\to\widetilde{\Cal{L}}$,
and hence also a mapping $\beta^{*}:\mathbb{W}(\Cal{L}\cup
M)\to\mathbb{W}(\widetilde{\Cal{L}})$. Assume finally:

###### Theorem 10.2

Assume Hypothesis 10.1. Then there exists a unique homomorphism
$\gamma:\Cal{L}^{+}\to\widetilde{\Cal{L}}$ such that
$\gamma\mid_{\Cal{L}}=\alpha$ and $\gamma\mid_{M}=\gamma_{M}$.

###### Demonstration Proof

Write $\Pi:\mathbb{D}\to\Cal{L}$, $\Pi^{+}:\mathbb{D}^{+}\to\Cal{L}^{+}$, and
$\widetilde{\Pi}:\widetilde{\mathbb{D}}\to\widetilde{\Cal{L}}$ for the
products on $\Cal{L}$, $\Cal{L}^{+}$, and $\widetilde{\Cal{L}}$, respectively.

By 3.1, $\Cal{F}$ is $\Delta^{+}$-inductive. Thus, for any $P\in R^{\Cal{F}}$
there exists $x_{P}\in\Cal{L}^{+}$ such that $N_{S}(P)\leq S_{x_{P}}$ and with
$P^{x_{P}}=R$. We have $P\in\Omega$ by 10.1(1), so $P<N_{S}(P)$, and then
$N_{S}(P)\in\Delta$ since $\Delta^{+}=\Delta\cup R^{\Cal{F}}$. Thus
$x_{P}\in\Cal{L}$ for all $P\in R^{\Cal{F}}$. In choosing the elements
$x_{P}$, we take $x_{R}=\mathbb{1}$.

Let $f\in\Cal{L}^{+}$ and assume that there exists $P\in R^{\Cal{F}}$ with
$P\leq S_{f}$. Set $Q=P^{f}$ and set
$u=(x_{P},x_{P}^{-1},f,x_{Q},x_{Q}^{-1})$. Then $u\in\mathbb{D}^{+}$ via $P$,
and we have

$f=\Pi^{+}(u)=x_{P}gx_{Q}^{-1},$

where $g=\Pi^{+}(x_{P}^{-1},f,x_{Q})\in M$. Then
$((x_{P})\alpha,g\gamma_{M},(x_{Q}^{-1})\alpha)\in\widetilde{\mathbb{D}}$ by
10.1(3). Set

If $P<S_{f}$ then $S_{f}\in\Delta$ and $u\in\mathbb{D}$ via $N_{S_{f}}(P)$. In
that case we have

$f\gamma=\widetilde{\Pi}((x_{P})\alpha,g\alpha,(x_{Q}^{-1})\alpha)=\widetilde{\Pi}(u\alpha^{*})=\Pi(u)\alpha,$

as $\alpha:\Cal{L}\to\widetilde{\Cal{L}}$ is a homomorphism. Thus
$f\gamma=f\alpha$ if $S_{f}$ properly contains a conjugate of $R$. For every
element $f\in\Cal{L}$ we have $S_{f}\in\Delta^{+}$, so either $f\in\Cal{L}$ or
$S_{f}$ contains a conjugate of $R$. We may therefore extend $\gamma$ to a
mapping $\Cal{L}^{+}\to\widetilde{\Cal{L}}$ by taking
$\gamma\mid_{\Cal{L}}=\alpha$. If $f\in M$ then we may take $P=Q=R$, and since
$x_{R}=\mathbb{1}$ we obtain $f\gamma=f\gamma_{M}$ from (4). Thus, we now have
$\gamma\mid_{\Cal{L}}=\alpha$ and $\gamma\mid_{M}=\gamma_{M}$.

We next show that $\gamma$ is a homomorphism. That is, we will show:

Let $w=(f_{1},\cdots,f_{n})\in\mathbb{D}^{+}$ be given. If $S_{w}\in\Delta$
then $w\in\mathbb{D}$, $w\gamma^{*}=w\alpha^{*}$, and then (5) holds since
$\alpha$ is a homomorphism. On the other hand, suppose
$S_{w}\notin\mathbb{D}$, so that $S_{w}\in R^{\Cal{F}}$. Set $P_{0}=S_{w}$ and
recursively define $P_{i}$ for $1\leq i\leq n$ by $P_{i}=(P_{i-1})^{f_{i}}$.
For $0\leq i\leq n$ set $x_{i}=x_{P_{i}}$, and for $1\leq i\leq n$ set
$u_{i}=(x_{i-1}^{-1},f_{i},x_{i})$. Noting that $u_{i}\in\mathbb{D}^{+}$ via
$P_{i-1}$, set $g_{i}=\Pi^{+}(u_{i})$, and set
$v_{i}=(x_{i-1},g_{i},x_{i}^{-1})$. Thus $g_{i}\in M$ and
$\Pi^{+}(v_{i})=f_{i}$. Set $v=v_{1}\circ\cdots\circ v_{n}$. Then
$v\in\mathbb{D}^{+}$ via $P_{0}$, and $v\in\mathbb{W}(\Cal{L}\cup M)$. Then
10.1(3) yields $v\gamma^{*}\in\widetilde{\mathbb{D}}$. An appeal to
$\widetilde{\mathbb{D}}$-associtivity then yields

$(\widetilde{\Pi}(v_{1}\gamma^{*})\circ\cdots\circ\widetilde{\Pi}(v_{n}\gamma^{*}))\in\widetilde{\mathbb{D}}.$
$None$

Here $\widetilde{\Pi}(v_{i}\gamma^{*})=(\Pi^{+}(v_{i}))\gamma$ by (4), and
$\Pi^{+}(v_{i})=f_{i}$. Thus (6) may be written as:
$w\gamma^{*}\in\widetilde{\mathbb{D}}$.

We have $\widetilde{\Pi}(w\gamma^{*})=\widetilde{\Pi}(v\gamma^{*})$, as we
have seen via $\widetilde{\mathbb{D}}$-associativity. As
$\gamma\mid_{\Cal{L}}$ is a homomorphism, a further application of
$\widetilde{\mathbb{D}}$-associativity yields

$\widetilde{\Pi}(w\gamma^{*})=\widetilde{\Pi}(x_{0}\gamma,g_{1}\gamma,\cdots,g_{n}\gamma,x_{n}^{-1}\gamma).$

Set $g=\Pi^{+}(g_{1},\cdots,g_{n})$. As $\gamma\mid_{M}$ is a homomorphism we
obtain

$\widetilde{\Pi}(w\gamma^{*})=\widetilde{\Pi}(x_{0}\gamma,g\gamma,x_{n}^{-1})=\widetilde{\Pi}(x_{0}\alpha,g\gamma_{M},x_{n}^{-1}),$

and so $\widetilde{\Pi}(w\gamma^{*})=\Pi^{+}(x_{1},g,x_{n}^{-1})\gamma$ by
(4). Since $\Pi^{+}(x_{1},g,x_{n}^{-1})=\Pi^{+}(v)=\Pi^{+}(w)$ by
$\mathbb{D}^{+}$-associativity, we have established (5). That is, $\gamma$ is
a homomorphism. We have seen that every $f\in\Cal{L}^{+}$ is a product
$\Pi^{+}(u)$ with $u\in\mathbb{W}(\Cal{L}\cup M)$, so $\Cal{L}^{+}$ is
generated as a partial group (I.1.9) by $\Cal{L}\cup M$. This observation
establishes the uniqueness of $\gamma$, and completes the proof. ∎

###### Definition Definition 10.3

Let $\Cal{L}$ be a partial group, and let $X$ and $Y$ be subsets of $\Cal{L}$.
Then $X$ is $Y$-invariant (or $X$ is invariant under $Y$) if for each $x\in X$
and each $y\in Y$ we have $(y^{-1},x,y)\in\mathbb{D}(\Cal{L})$ and $x^{y}\in
Y$.

###### Corollary 10.4

Let $(\Cal{L},\Delta,S)$ be a proper locality on $\Cal{F}$, and let
$(\Cal{L}^{\prime},\Delta^{\prime},S)$ be an expansion of $\Cal{L}$ to a
proper locality on $\Cal{F}$. Let
$(\widetilde{\Cal{L}},\widetilde{\Delta},\widetilde{S})$ be a locality, and
let $\alpha:\Cal{L}\to\widetilde{\Cal{L}}$ be a homomorphism. Assume that
there exists a subgroup $\widetilde{V}\leq\widetilde{S}$ such that
$\widetilde{V}$ is invariant under $Im(\alpha)$ and such that:

$\\{(P\alpha)\widetilde{V}\mid
P\in\Delta^{\prime}\\}\subseteq\widetilde{\Delta}.$ $None$

Then $\alpha$ extends in a unique way to a homomorphism
$\alpha^{\prime}\Cal{L}^{\prime}\to\widetilde{\Cal{L}}$.

###### Demonstration Proof

Set $\Omega=\Omega_{S}(\Cal{L})$, and let $(*,\Omega)$ be the stratification
on $\Cal{F}$ induced from $\Cal{L}^{\prime}$. For $X\leq S$ write $dim(X)$ for
$dim_{\Omega}(X)$, and let $m$ be the maximum, taken over all
$X\in\Delta^{\prime}-\Delta$, of $dim(X)$. Among all counter-examples to 10.4,
assume that $(\Cal{L},\Cal{L}^{\prime},\alpha)$ has been chosen so that $m$ is
as small as possible.

Let $\Sigma$ be the set of all subgroups $Y\leq S$ such that $Y^{*}\in\Delta$.
Then $\Delta\subseteq\Sigma$, and $\Sigma$ is a set of objects for $\Cal{L}$
by I.3.16. Similarly, the set $\Sigma^{\prime}$ of all $Y\leq S$ such that
$Y^{*}\in\Delta^{\prime}$ is a set of objects for $\Cal{L}^{\prime}$, and we
have $\Sigma\subseteq\Sigma^{\prime}$. We may therefore take $\Delta=\Sigma$
and $\Delta^{\prime}=\Sigma^{\prime}$. Thus, we have reduced to the case where
$X\in\Delta$ for all subgroups $X\leq S$ such that $X^{*}\in\Delta$.

Choose $R\in\Delta^{\prime}$ with $R\notin\Delta$, and so that $dim(R)=m$.
Then $dim(R^{*})=m$ (as otherwise $R\in\Delta$), and thus $R\in\Omega$. As
$dim(R)=dim(P)$ for $P\in R^{\Cal{F}}$, we may take $R$ to be fully normalized
in $\Cal{F}$ relative to $(*,\Omega)$. For any $P\in R^{\Cal{F}}$ and any
subgroup $X$ of $S$ with $P<S$ we have $dim(X)>m$, and so $X\in\Delta$. Thus
the set $\Delta^{+}:=\Delta\cup R^{\Cal{F}}$ is $\Cal{F}$-closed. Let
$(\Cal{L}^{+},\Delta^{+},S)$ be the restriction of $\Cal{L}^{\prime}$ to
$\Delta^{+}$. We now have the setup of 10.1(1).

Set $M=N_{\Cal{L}^{+}}(R)$. Then $M$ is a subgroup of $\Cal{L}$, by 7.1. Thus
10.1(2) obtains. In order to verify 10.1(3), take $w\in\mathbb{W}(\Cal{L})$,
set $P=S_{w}$, and assume $P\in\Delta^{+}$. Then $P\in\Delta^{\prime}$, and so
$P\alpha\widetilde{V}\in\widetilde{\Delta}$ by (*). As $\widetilde{V}$ is
$Im(\alpha)$-invariant, we have $P\alpha\widetilde{V}\leq S_{w\alpha^{*}}$,
and thus $w\alpha^{*}\in\mathbb{D}(\widetilde{\Cal{L}})$. That is, 10.1(3)
holds. Now 10.2 yields a unique extension of $\alpha$ to a homomorphism
$\gamma:\Cal{L}^{+}\to\widetilde{\Cal{L}}$.

By I.2.3(c) $\widetilde{V}$ is invariant under all conjugation maps $c_{u}$,
where $u\in\mathbb{W}(Im(\alpha)$. As $\Cal{L}^{+}$ is generated as a partial
group by $\Cal{L}$ it follows that $\widetilde{V}$ is invariant under
$Im(\gamma)$. We therefore have the hypothesis of 10.4 fulfilled with
$\Cal{L}^{+}$ and $\gamma$ in place of $\Cal{L}$ and $\alpha$. As
$dim(R)<dim(X)$ for $X\in\Delta$, we now obtain the desired conclusion via the
minimality of $m$. ∎

## References

  * 5a C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver, Subgroup families controlling $p$-local finite groups, Proc. London Math. Soc. 91 (2005), 325-354.
  * AKO Michael Aschbacher, Radha Kessar, and Bob Oliver, Fusion systems in algebra and topology, London Mathematical Society Lecture Note Series, vol. 391, Cambridge University Press, Cambridge, 2011.
  * Ch1 Andrew Chermak, Fusion systems and localities, Acta Math. 211 (2013), 47-139.
  * Ch2 —, Finite localities I, (arXiv:1505.07786) (2021).
  * Ch3 —, Finite localities II , (arXiv:1505.08110) (2021).
  * CG Andrew Chermak and Alex Gonzales, Discrete localities I, (arXiv:) (2021).
  * Cr David Craven, The theory of fusion systems: An algebraic appoach, Cambridge Studies in Advanced Mathematics, vol. . 131, Cambridge University Press, Cambridge, 2011.
  * Gold David Goldschnidt, A conjugation family for finite groups, J.Algebra 16 (1970), 138-142.
  * Gor Daniel Gorenstein, Finite groups, Second Edition, Chelsea, New York, 1980.
  * He1 Ellen Henke, Products of partial normal subgroups, Pacific J. Math. 279 (2015), 255-268.
  * He2 —, Subcentric linking systems, Trans. Amer. Math. Soc. 371 (2019), 3325-3373.
  * He3 —, Extensions of homomorphisms between localities, arXiv:2019.12626 (2021).
  * LL Ran Levi and Assaf Libman, Existence and uniqueness of classifying spaces for fusion systems over discrete $p$-toral groups, J. of London Math. Soc. 91 (2015), 47-70.
  * M Remi Molinier, Control of fixed points over discrete $p$-toral groups, and existence and uniqueness of linking systems, J. of Alg. 499 (2018), 43-73.