EXTENSIONS OF THE BENSON-SOLOMON FUSION SYSTEMS

arXiv:1701.06949v1 [] 24 Jan 2017

ELLEN HENKE AND JUSTIN LYND
To Dave Benson on the occasion of his second 60th birthday
Abstract. The Benson-Solomon systems comprise the only known family of simple saturated
fusion systems at the prime two that do not arise as the fusion system of any ﬁnite group. We
determine the automorphism groups and the possible almost simple extensions of these systems
and of their centric linking systems.

1. Introduction
b is controlled
b with normal subgroup G such that C b (G) 6 G, the structure of G
Given a group G
G
by G and Aut(G). On the other hand, if Fb is a saturated fusion system with normal subsystem
F such that CFb (F) 6 F, it is in general more difficult to describe the possibilities for Fb given F.
For example, if F = FS (G) is the fusion system of the finite group G, then it is often the case
that some extensions of F do not arise as extensions of G, but rather as extensions of a different
finite group with the same fusion system. It is also possible that there are extensions of FS (G)
that are exotic in the sense that they are induced by no finite group, but to our knowledge, no
example of this is currently known.
It has become clear that working with the fusion system alone is not on-its-face enough for
describing extensions [Oli10, AOV12, Oli16, BMO16]; one should instead work with an associated
linking system. In particular, when F = FS (G) and L = LcS (G) for some finite group G, there is a
natural map κG : Out(G) → Out(L), and whether the extensions of F all come from finite groups
has been shown to be related to asking whether κG′ is split surjective for some other finite group
G′ with the same Sylow subgroup and with F = FS (G′ ) [AOV12, Oli16]. While this machinery is
not directly applicable to the problem of determining extensions of exotic fusion systems, there
are other tools for use, such as [Oli10].
There is one known family of simple exotic fusion systems at the prime 2, the Benson-Solomon
systems. They were first predicted by Dave Benson [Ben98] to exist as finite versions of a 2local compact group associated to the 2-compact group DI(4) of Dwyer and Wilkerson [DW93].
They were later constructed by Levi and Oliver [LO02] and Aschbacher and Chermak [AC10].
The purpose of this paper is determine the automorphism groups of the Benson-Solomon fusion
and centric linking systems, and use that information to determine the fusion systems having
one of these as its generalized Fitting subsystem. This information is needed within certain
portions of Aschbacher’s program to classify simple fusion systems of component type at the
prime 2. In particular, it is presumably needed within an involution centralizer problem for
Date: April 4, 2018.
2000 Mathematics Subject Classification. Primary 20D20, Secondary 20D05.
Key words and phrases. fusion system, linking system, Benson-Solomon fusion system, group extension.
Justin Lynd was partially supported by NSA Young Investigator Grant H98230-14-1-0312. This project has
received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie
Sklodowska-Curie grant agreement No. 707758.
1

these systems. Some of the work on automorphisms of these systems appears in the standard
references [LO02, LO05, AC10], and part of our aim is to complete the picture.
The plan for this paper is as follows. All maps are written on the left. In Section 2, we
recall the various automorphism groups of fusion and linking systems and the maps between
them, following [AOV12]. In Section 3, we look at automorphisms of the fusion and linking
systems of Spin7 (q) and of the Benson-Solomon systems. We show in Theorem 3.10 that the
outer automorphism group of the latter is a cyclic group of field automorphisms of 2-power order.
Finally, we show in Theorem 4.3 that the systems having a Benson-Solomon generalized Fitting
subsystem are uniquely determined by the outer automorphisms they induce on the fusion system,
and that all such extensions are split.
We would like to thank Jesper Grodal, Ran Levi, and Bob Oliver for helpful conversations.
2. Automorphisms of fusion and linking systems
We refer to [AKO11] for the definition of a saturated fusion system, and also for the definition
of a centric subgroup of a fusion system. Let F be a saturated fusion system over the finite
p-group S, and write F c for the collection of F-centric subgroups. Whenever g is an element of a
finite group, we write cg for the conjugation homomorphism x 7→ gx = gxg −1 and its restrictions.
2.1. Background on linking systems. Whenever ∆ is an overgroup-closed, F-invariant collection of subgroups of S, we have the transporter category T∆ (S) with those objects. This is
the full subcategory of the transporter category TS (S) where the objects are subgroups of S, and
morphisms are the transporter sets: NS (P, Q) = {s ∈ S | sP s−1 6 Q} with composition given by
multiplication in S.
A linking system associated to F is a nonempty category L with object set ∆, together with
functors
(2.1)

δ

π

T∆ (S) −−−−→ L −−−−→ F.

The functor δ is the identity on objects and injective on morphisms, while π is the inclusion on
objects and surjective on morphisms. Write δP,Q for the corresponding injection NS (P, Q) →
MorL (P, Q) on morphisms, write δP for δP,P , and use similar notation for π.
The category and its structural functors are subject to several axioms which may be found
in [AKO11, Definition II.4.1]. In particular, Axiom (B) states that for all objects P and Q in L
and each g ∈ NS (P, Q), we have πP,Q(δP,Q (g)) = cg ∈ HomF (P, Q). A centric linking system is
a linking system with ∆ = F c . Given a finite group G with Sylow p-subgroup S, the canonical
centric linking system for G is the category LcS (G) with objects the p-centric subgroups P 6 S
(namely those P whose centralizer satisfies CG (P ) = Z(P ) × Op′ (CG (P ))), and with morphisms
the orbits of the transporter set NG (P, Q) = {g ∈ G | gP g−1 6 Q} under the right action of
Op′ (CG (P )).
2.1.1. Distinguished subgroups and inclusion morphisms. The subgroups δP (P ) 6 AutL (P ) are
called distinguished subgroups. When P 6 Q, the morphism ιP,Q := δP,Q (1) ∈ MorL (P, Q) is the
inclusion of P into Q.
2.1.2. Axiom (C) for a linking system. We will make use of Axiom (C) for a linking system, which
says that for each morphism ϕ ∈ MorL (P, Q) and element g ∈ NS (P ), the following identity holds
between morphisms in MorL (P, Q):
ϕ ◦ δP (g) = δQ (π(ϕ)(g)) ◦ ϕ.
2

2.1.3. Restrictions in linking systems. For each morphism ψ ∈ MorL (P, Q), and each P0 , Q0 ∈
Ob(L) such that P0 6 P , Q0 6 Q, and π(ψ)(P0 ) 6 Q0 , there is a unique morphism ψ|P0 ,Q0 ∈
MorL (P0 , Q0 ) (the restriction of ψ) such that ψ ◦ ιP0 ,P = ιQ0 ,Q ◦ ψ|P0 ,Q0 ; see [Oli10, Proposition 4(b)] or [AKO11, Proposition 4.3].
Note that in case ψ = δP,Q (s) for some s ∈ NS (P, Q), it can be seen from Axioms (B) and (C)
that ψ|P0 ,Q0 = δP0 ,Q0 (s).
2.2. Background on automorphisms.
2.2.1. Automorphisms of fusion systems. An automorphism of F is, by definition, determined by
its effect on S: define Aut(F) to be the subgroup of Aut(S) consisting of those automorphisms α
αϕα−1

which preserve fusion in F in the sense that the homomorphism given by α(P ) −−−−→ α(Q) is in
ϕ
F for each morphism P −
→ Q in F. The automorphisms AutF (S) of S in F thus form a normal
subgroup of Aut(F), and the quotient Aut(F)/ AutF (S) is denoted by Out(F).
2.2.2. Automorphisms of linking systems. A self-equivalence of L is said to be isotypical if it
sends distinguished subgroups to distinguished subgroups, i.e. α(δP (P )) = δα(P ) (α(P )) for each
object P . It sends inclusions to inclusions provided α(ιP,Q ) = ια(P ),α(Q) whenever P 6 Q. The
monoid Aut(L) of isotypical self-equivalences that send inclusions to inclusions is in fact a group
of automorphisms of the category L, and this has been shown to be the most appropriate group of
automorphisms to consider. Note that Aut(L) has been denoted by AutItyp (L) in [AKO11,AOV12]
and elsewhere. When α ∈ Aut(L) and P is an object with α(P ) = P , we denote by αP the
automorphism of AutL (P ) induced by α.
The group AutL (S) acts by conjugation on L in the following way: given γ ∈ AutL (S), consider
the functor cγ ∈ Aut(L) which is cγ (P ) = π(γ)(P ) on objects, and which sends a morphism
ϕ
P −
→ Q in L to the morphism γϕγ −1 from cγ (P ) to cγ (Q) after replacing γ and γ −1 by the
appropriate restrictions (introduced in §2.1.3). Note that when γ = δS (s) for some s ∈ S, then
cγ (P ) is conjugation by s on objects, and cγ (ϕ) = δQ,sQ (s) ◦ ϕ ◦ δsP,P (s−1 ) for each morphism ϕ ∈
MorL (P, Q) by the remark on distinguished morphisms in §2.1.3. In particular, when L = LcS (G)
for some finite group G, cγ is truly just conjugation by s on morphisms.
The image of AutL (S) under the map γ 7→ cγ is seen to be a normal subgroup of Aut(L); the
outer automorphism group of L is
Out(L) := Aut(L)/{cγ | γ ∈ AutL (S)};
we refer to Lemma 1.14(a) and the surrounding discussion in [AOV12] for more details. This
group is denoted by Outtyp (L) in [AKO11, AOV12] and elsewhere.
2.2.3. From linking system automorphisms to fusion system automorphisms. There is a group
homomorphism
(2.2)

µ
e : Aut(L) −→ Aut(F),

given by restriction to S ∼
e induces a
= δS (S) 6 AutL (S); see [Oli10, Proposition 6]. The map µ
homomorphism on quotient groups
µ : Out(L) −→ Out(F).
We write µL (or µG when L = LcS (G)) whenever we wish to make clear which linking system we
are working with; similar remarks hold for µ
e. As shown in [AKO11, Proposition II.5.12], ker(µ)
has an interesting cohomological interpretation as the first cohomology group of the center functor
3

ZF on the orbit category of F-centric subgroups, and ker(e
µ) is correspondingly a certain group
of normalized 1-cocycles for this functor.
2.2.4. From group automorphisms to fusion system and linking system automorphisms. We also
need to compare automorphisms of groups with the automorphisms of their fusion and linking
systems. If G is a finite group with Sylow p-subgroup S, then each outer automorphism of G is
represented by an automorphism that fixes S. This is a consequence of the transitive action of G
on its Sylow subgroups. More precisely, there is an exact sequence:
incl

g7→cg

1 → Z(G) −−→ NG (S) −−−→ Aut(G, S) → Out(G) → 1.
where Aut(G, S) is the subgroup of Aut(G) consisting of those automorphisms that leave S
invariant.
For each pair of p-centric subgroups P, Q 6 S and each α ∈ Aut(G, S), α induces an isomorphism Op′ (CG (P )) → Op′ (CG (α(P ))) and a bijection NG (P, Q) → NG (α(P ), α(Q)). Thus, there
is a group homomorphism
κ
eG : Aut(G, S) → Aut(LcS (G))
which sends α ∈ Aut(G, S) to the functor which is α on objects, and also α on morphisms in
the way just mentioned. This map sends the image of NG (S) to {cγ | γ ∈ AutLcS (G) (S)}, and so
induces a homomorphism
κG : Out(G) → Out(LcS (G))
on outer automorphism groups.
It is straightforward to check that the restriction to S of any member of Aut(G, S) is an
automorphism of the fusion system FS (G). Indeed, for every α ∈ Aut(G, S), the automorphism
α|S of FS (G) is just the image of α under µ
eG ◦ κ
eG .
2.2.5. Summary. What we will need in our proofs is summarized in the following commutative
diagram, which is an augmented and updated version of the one found in [AKO11, p.186].

Z(F)

incl

1

1

1


// Z(S)


b1 (O(F c ), ZF )
// Z


// lim1 (ZF )

e
λ

δS

(2.3)

Z(F)



// AutL (S)



// Aut(L)

πS

1



// AutF (S)


1

←−

λ



// Out(L)

// Aut(F)


1

// 1

µ

µ
e



// 1



// Out(F)

// 1



1

All sequences in this diagram are exact. Most of this either is shown in the proof of [AKO11,
Proposition II.5.12], or follows from the above definitions. The first and second rows are exact by
this reference, except that the diagram was not augmented by the maps out of Z(F) (the center
of F); exactness at Z(S) and AutL (S) is shown by following the proof there. Given [AKO11,
Proposition II.5.12], exactness of the last column is equivalent to the uniqueness of centric linking
4

systems, a result of Chermak. In all the cases needed in this article, it follows from [LO02,
Lemma 3.2]. The second-to-last column is then exact by a diagram chase akin to that in a
5-lemma for groups.

3. Automorphisms
The isomorphism type of the fusion systems of the Benson-Solomon systems FSol (q), as q ranges
over odd prime powers, is dependent only on the 2-share of q 2 − 1 by [COS08, Theorem B]. Since
the centralizer of the center of the Sylow group is the fusion system of Spin7 (q), the same holds
also for the fusion systems of these groups. For this reason, and because of Proposition 3.2 below,
k
it will be convenient to take q = 52 for the sequel. Thus, we let F be the algebraic closure of the
field with five elements.
3.1. Automorphisms of the fusion system of Spin7 (q). Let H̄ = Spin7 (F). Fix a maximal
torus T̄ of H̄. Thus, H̄ is generated by the T̄ -root groups X̄α = {xα (λ) : λ ∈ F} ∼
= (F, +), as
α ranges over the root system of type B3 , and is subject to the Chevalley relations of [GLS98,
Theorem 1.12.1]. For any power q1 of 5, we let ψq1 denote the standard Frobenius endomorphism
of H̄, namely the endomorphism of H̄ which acts on the root groups via ψq1 (xα (λ)) = xα (λq1 ).
Set H := CH̄ (ψq ). Thus, H = Spin7 (q) since H̄ is of universal type (see [GLS98, Theorem 2.2.6(f)]). Also, T := CT̄ (ψq ) is a maximal torus of H. For each power q1 of 5, the Frobenius
endomorphism ψq1 of H̄ acts on H in the way just mentioned, and it also acts on T by raising
each element to the power q1 . For ease of notation, we denote by ψq1 also the automorphism of
H induced by ψq1 .
The normalizer NH (T ) contains a Sylow 2-subgroup H [GL83, 10-1(2)], and NH (T )/T is isomorphic to C2 × S4 , the Weyl group of B3 . Applying a Frattini argument, we see that there is a
Sylow 2-subgroup S of NH (T ) invariant under ψ5 , and we fix this choice of S for the remainder.
The automorphism groups of the Chevalley groups were determined by Steinberg [Ste60], and
in particular,
(3.1)

Out(H) = Outdiag(H) × Φ ∼
= C2 × C2k ,

where Φ is the group of field automorphisms, and where Outdiag(H) is the group of outer automorphisms of H induced by NT̄ (H) [GLS98, Theorem 2.5.1(b)]. We mention that S is normalized
by every element of NT̄ (H). So we find canonical representatives of the elements of Φ and of
Outdiag(H) in Aut(H, S).
We need to be able to compare automorphisms of the group with automorphisms of the fusion
and linking systems, and this has been carried out in full generality by Broto, Møller, and Oliver
[BMO16] for groups of Lie type.
Let FSpin (q) and LcSpin (q) be the associated fusion and centric linking systems over S of the
group H, and recall the maps µH and κH from §§2.2.3, 2.2.4
Proposition 3.2. The maps µH and κH are isomorphisms, and hence
k

k

Out(LSpin (52 )) ∼
= Out(FSpin (52 )) ∼
= C2 × C2k .
Proof. That µH is an isomorphism follows from (2.3) and [LO02, Lemma 3.2]. Also, κH is an
isomorphism by [BMO16, Propositions 5.14, 5.15].

5

3.2. Automorphisms of the Benson-Solomon systems. We keep the notation from the
previous subsection. We denote by F := FSol (q) a Benson-Solomon fusion system over the 2group S ∈ Syl2 (H) fixed above, and by L := LcSol (q) an associated centric linking system with
structural functors δ and π. Set
T2 := T ∩ S,
the 2-torsion in the maximal torus T of H. Then T2 is homocyclic of rank three and of exponent
2k+2 [AC10, §4]. Also, Z(S) 6 T2 is of order 2, and NF (Z(S)) = CF (Z(S)) = FSpin (q) is a fusion
system over S.
Since Z(S) is contained in every F-centric subgroup, by Definition 6.1 and Lemma 6.2 of
[BLO03], we may take NL (Z(S)) = CL (Z(S)) for the centric linking system of Spin7 (q). By the
items just referenced, CL (Z(S)) is a subcategory of L with the same objects, and with morphisms
those morphisms ϕ in L such that π(ϕ)(z) = z. Further, CL (Z(S)) has the same inclusion
functor δ, and the projection functor for CL (Z(S)) is the restriction of π. (This was also shown
in [LO02, Lemma 3.3(a,b)].)
Write Fz for FSpin (q) and Lz for CL (Z(S)) for short. Each member of Aut(F) fixes Z(S) and
so Aut(F) ⊆ Aut(Fz ). So the inclusion map from Aut(F) to Aut(Fz ) can be thought of as a
“restriction map”
(3.3)

ρ : Aut(F) −→ Aut(Fz )

given by remembering only that an automorphism preserves fusion in Fz . We want to make
explicit in Lemma 3.5 that the map ρ of (3.3) comes from a restriction map on the level of centric
linking systems. First we need to recall some information about the normalizer of T2 in L and
Lz .
Lemma 3.4. The following hold after identifying T2 with its image δT2 (T2 ) 6 AutL (T2 ).
(a) AutLz (T2 ) is an extension of T2 by C2 × S4 , and AutL (T2 ) is an extension of T2 by
C2 × GL3 (2) in which the GL3 (2) factor acts naturally on T2 /Φ(T2 ). In each case, a C2
factor acts as inversion on T2 . Also, T2 is equal to its centralizer in each of the above
normalizers, Z(AutLz (T2 )) = Z(S), and Z(AutL (T2 )) = 1.
(b) AutF (S) = Inn(S) = AutFz (S) and AutL (S) = δS (S) = AutLz (S).
Proof. For part (a), see Lemma 4.3 and Proposition 5.4 of [AC10].
Since T2 is the unique abelian subgroup of its order in S by [AC10, Lemma 4.9(c)], it is
characteristic. By the uniqueness of restrictions 2.1.3, we may therefore view AutL (S) as a
subgroup of AutL (T2 ). Since AutL (T2 ) has self-normalizing Sylow 2-subgroups by (a), the same
holds for AutL (S). Now (b) follows for L, and for F after applying π. This also implies the
statement for Lz and Fz , as subcategories.

There is a 3-dimensional commutative diagram related to (2.3) that is the point of the next
lemma.
Lemma 3.5. There is a restriction map ρb: Aut(L) → Aut(Lz ), with kernel the automorphisms
induced by conjugation by δS (Z(S)) 6 AutL (S), which makes the diagram
Aut(L)

ρb

// Aut(Lz )
µ
eLz

µ
eL





Aut(F)

ρ
6

// Aut(Fz ) ,

commutative, which commutes with the conjugation maps out of
AutL (S)

id

// AutL (S)
z

πS

πS



AutF (S)



id

// AutF (S),
z

and which therefore induces a commutative diagram
Out(L)

[b
ρ]

// Out(Lz )
µLz

µL



Out(F)



[ρ]

// Out(Fz ).

Proof. Recall that we have arranged Lz ⊆ L. Thus, the horizontal maps in the second diagram
are the identity maps by Lemma 3.4, and so the lemma amounts to checking that an element of
Aut(L) sends morphisms in Lz to morphisms in Lz . For then, we can define its image under ρb to
have the same effect on objects, and to be the restriction to Lz on morphisms.
Now fix an arbitrary α ∈ Aut(L), objects P, Q ∈ F c = Fzc , and a morphism ϕ ∈ MorL (P, Q).
Let Z(S) = hzi. By two applications of Axiom (C) for a linking system (§§2.1.2),
(3.6)

ιP,S ◦ δP (z) = δS (z) ◦ ιP,S

and

ια(P ),S ◦ δα(P ) (z) = δS (z) ◦ ια(P ),S ,

because π(ιP,S )(z) = π(ια(P ),S )(z) = z. Since αS is an automorphism of δS (S) ∼
= S, it sends δS (z)
to itself. Thus, α sends the right side of the first equation of (3.6) to the right side of the second,
since it sends inclusions to inclusions. Thus
ια(P ),S ◦ α(δP (z)) = ια(P ),S ◦ δα(P ) (z).
However, each morphism in L is a monomorphism [Oli10, Proposition 4], so we obtain
(3.7)

α(δP (z)) = δα(P ) (z),

and the same holds for Q in place of P .
Since ϕ ∈ Mor(Lz ), we have π(ϕ)(z) = z, so by two more applications of Axiom (C),
(3.8)

ϕ ◦ δP (z) = δQ (z) ◦ ϕ

and

α(ϕ) ◦ δα(P ) (z) = δα(Q) (π(α(ϕ))(z)) ◦ α(ϕ).

After applying α to the left side of the first equation of (3.8), we obtain the left side of the second
by (3.7). Thus, comparing right sides, we obtain
δα(Q) (z) ◦ α(ϕ) = δα(Q) (π(α(ϕ)(z))) ◦ α(ϕ)
Since each morphism in L is an epimorphism [Oli10, Proposition 4], it follows that
δα(Q) (z) = δα(Q) (π(α(ϕ))(z)).
Hence, π(α(ϕ))(z) = z because δα(Q) is injective (Axiom (A2)). That is, α(ϕ) ∈ Mor(Lz ) as
required.
The kernel of ρb is described via a diagram chase in (2.3). Suppose ρb(α) is the identity. Then,
α is sent to the identity automorphism of S by µ
eL , since ρ is injective. Thus, α comes from a
normalized 1-cocycle by (2.3) and these are in turn induced by elements of Z(S) since lim1 (ZF )
←−
is trivial [LO02, Lemma 3.2].

7

Lemma 3.9. Let G be a finite group and let V be an abelian normal 2-subgroup of G such
that CG (V ) 6 V . Let α be an automorphism of G such that [V, α] = 1 and α2 ∈ Inn(G). Then
[G, α] 6 V , and if G acts fixed point freely on V /Φ(V ), then the order of α is at most the exponent
of V .
Proof. As [V, α] = 1, we have [V, G, α] 6 [V, α] = 1 and [α, V, G] = [1, G] = 1. Hence, by the
Three subgroups lemma, it follows [G, α, V ] = 1. As CG (V ) 6 V , this means
[G, α] 6 V.
Assume from now on that G acts fixed point freely on V /Φ(V ). Write G∗ := G ⋊ hαi for the
semidirect product of G by hαi. As [V, α] = 1 and [G, α] 6 V , the subgroup W := V hαi is an
abelian normal subgroup of G∗ with [W, G∗ ] 6 V .
As [V, α] = 1, it follows [V, α2 ] = 1. So α2 ∈ Inn(G) is realized by conjugation with an element
−1 2
of CG (V ) = V . Pick u ∈ V with α2 = cu |G . This means that, for any g ∈ G, we have u α g = g
in G∗ . So Z := hu−1 α2 i centralizes G in G∗ . Since W is abelian and contains Z, it follows that
Z lies in the centre of G∗ = W G. Set
G∗ = G∗ /Z.
Because CG (V ) 6 V , the order of u equals the order of cu |G = α2 . Hence, Z ∩ G = 1 = Z ∩ hαi.
So |ᾱ| = |α| and G ∼
= Ḡ. In particular, we have V̄ ∼
= V and Ḡ acts fixed point freely on
2
V̄ /Φ(V̄ ). Note also that ᾱ = ū. Hence |W̄ /V̄ | = 2 and Φ(W̄ ) 6 V̄ . Moreover, letting n ∈ N
such that 2n is the exponent of V , we have |α| = |ᾱ| 6 2 · 2n = 2n+1 . Assume |ᾱ| = 2n+1 .
Then ū = ᾱ2 has order 2n and is thus not a square in V̄ . Note that Φ(V̄ ) = {v 2 : v ∈ V̄ }
and Φ(W̄ ) = {w2 : w ∈ W } = hᾱ2 iΦ(V̄ ) 6 V̄ . Hence, Φ(W̄ )/Φ(V̄ ) has order 2. As Ḡ normalizes
Φ(W̄ )/Φ(V̄ ), it thus centralizes Φ(W̄ )/Φ(V̄ ) contradicting the assumption that Ḡ acts fixed point
freely on V̄ /Φ(V̄ ). Thus |α| = |ᾱ| 6 2n which shows the assertion.

We are now in a position to determine the automorphisms of F = FSol(q) and L = LcSol (q). It
is known that the field automorphisms induce automorphisms of these systems as we will make
precise next. Recall that the field automorphism ψ5 of H of order 2k normalizes S and so ψ5 |S
is an automorphism of Fz = FS (H). By [AC10, Lemma 5.7], the automorphism ψ5 |S is actually
also an automorphism of F. We thus denote it by ψF and refer to it as the field automorphism
of F induced by ψ5 . By Proposition 3.2, this automorphism has order 2k .
By [LO02, Proposition 3.3(d)], there is a unique lift ψ of ψF under µ
eL that is the identity on
π −1 (FSol (5)) and restricts to κ
eH (ψ5 ) on Lz . We refer to ψ as the field automorphism of L induced
by ψ5 .
Theorem 3.10. The map µL : Out(LcSol (q)) → Out(FSol (q)) is an isomorphism, and
Out(Lc (q)) ∼
= Out(FSol (q)) ∼
=C k
2

Sol

is induced by field automorphisms. Also, the automorphism group Aut(LcSol (q)) is a split extension
of S by Out(LcSol (q)); in particular, it is a 2-group.
More precisely, if ψ is the field automorphism of LcSol (q) induced by ψ5 , then ψ has order 2k
and Aut(LcSol (q)) is the semidirect product of AutL (S) ∼
= S with the cyclic group generated by ψ.
Proof. We continue to write L = LcSol (q), F = FSol (q), Lz = LcSpin (q), and Fz = FSpin (q), and we
continue to assume that L has been chosen so as to contain Lz as a linking subsystem. Recall
that T2 6 S is homocyclic of rank 3 and exponent 2k+2 .
We first check whether the outer automorphism of Lz induced by a diagonal automorphism
of H extends to L, and we claim that it doesn’t. A non-inner diagonal automorphism of H
8

is induced by conjugation by an element t of T̄ by [GLS98, Theorem 2.5.1(b)]. Its class as
an outer automorphism has order 2, so if necessary we replace t by an odd power and assume
that t2 ∈ T2 . Now T2 consists of the elements of T̄ of order at most 2k+2 , so t has order 2k+3
and induces an automorphism of H of order at least 2k+2 , depending on whether it powers into
Z(H) = Z(S) or not (in fact it does, but this is not needed). For ease of notation, we identify T2
with δT2 (T2 ) 6 AutL (T2 ), and we identify s ∈ S with δS (s) ∈ AutL (S).
Let τ = κ
eH ([ct ]) ∈ Aut(Lz ), and assume that τ lifts to an element τb ∈ Aut(L) under the
map ρb of Lemma 3.5. As ρb(b
τ) = τ = κ
eH (ct ), we have ρb(b
τ 2) = τ 2 = κ
eH (ct2 ), i.e. ρb(b
τ 2 ) acts on
every object and every morphism of Lz = LcS (H) as conjugation by t2 . Similarly, if we take the
conjugation automorphism ct2 of L by t2 (or more precisely the conjugation automorphism cδS (t2 )
of L by δS (t2 )), then ρb(ct2 ) is just the conjugation automorphism of Lz by t2 . So according to
the remark at the end of §§2.2.2, the automorphism ρb(ct2 ) acts also on Lz via conjugation by t2 ,
showing ρb(b
τ 2 ) = ρb(ct2 ). By the description of the kernel in Lemma 3.5, we have thus τb2 = ct2 or
2
τb = ct2 z .
Now set α := τbT2 ∈ Aut(AutL (T2 )). From what we have shown, it follows that α equals the
conjugation automorphism ct2 or ct2 z of AutL (T2 ). Note that |ct2 z | = |t2 z| = |t2 | = |ct2 |, since
Z(AutL (T2 )) = 1 by Lemma 3.4(a). Hence,
(3.11)

|α| = 2|α2 | = 2|t2 | = 2k+3 ,

On the other hand, α centralizes T2 , and we have seen that α2 is an inner automorphism of
AutL (T2 ). Moreover, by Lemma 3.4(a), CAutL (T2 ) (T2 ) = T2 , and AutL (T2 ) acts fixed point freely
on T2 /Φ(T2 ). The hypotheses of Lemma 3.9 thus hold for G = AutL (T2 ) and α ∈ Aut(G). So
α has order at most 2k+2 by that lemma, contradicting (3.11). We conclude that a diagonal
automorphism of Lz does not extend to an automorphism of L.
The existence of the field automorphism ψF of F, and the fact that ψF has order 2k , now yields
together with Proposition 3.2 that Out(F) ∼
= C2k is generated by the image of ψF in Out(F).
Moreover, by [LO02, Lemma 3.2] and the exactness of the third column of (2.3), the maps µL
and µLz are isomorphisms. Thus,
Out(L) ∼
= Out(F) ∼
= C k.
2

Let ψ be the field automorphism of L induced by ψ5 as above. Then ψ is the identity on
π −1 (FSol (5)) by definition. It remains to show that ψ has order 2k , since this will imply that
Aut(L) is a split extension of AutL (S) ∼
= S by hψi ∼
= Out(L) ∼
= Out(F).
k
2
The automorphism ψ maps to the trivial automorphism of F, and so is conjugation by an elek
/ Z(AutLcSol (5) (Ω2 (T2 )))
ment of Z(S) by (2.3). Now ψ 2 is trivial on AutLcSol (5) (Ω2 (T2 )), whereas z ∈
by Lemma 3.4(a) as Ω2 (T2 ) is the torus of LcSol (5). Thus, since a morphism ϕ is fixed by cz if and
k
only if π(ϕ)(z) = z (Axiom (C)), we conclude that ψ 2 is the identity automorphism of L, and
this completes the proof.

4. Extensions
In this section, we recall a result of Linckelmann on the Schur multipliers of the BensonSolomon systems, and we prove that each saturated fusion system F with F ∗ (F) ∼
= FSol (q) is a
split extension of F ∗ (F) by a group of outer automorphisms.
Recall that the hyperfocal subgroup of a saturated p-fusion system F over S is defined to be
the subgroup of S given by
hyp(F) = h[ϕ, s] := ϕ(s)s−1 | s ∈ P 6 S and ϕ ∈ O p (AutF (P ))i.
9

A subsystem F0 over S0 6 S is said to be of p-power index in F if hyp(F) 6 S0 and O p (AutF (P )) 6
AutF0 (P ) for each P 6 S0 . There is always a unique normal saturated subsystem on hyp(F) of
p-power index in F, which is denoted by Op (F) [AKO11, §I.7]. We will need the next lemma in
§§4.2.
Lemma 4.1. Let F be a saturated fusion system over S, and let F0 be a weakly normal subsystem
of F over S0 6 S. Assume that Op (AutF (S0 )) 6 AutF0 (S0 ). Then Op (AutF (P )) 6 AutF0 (P )
for every P 6 S0 . Thus, if in addition hyp(F) 6 S0 , then F0 has p-power index in F.
Proof. Note that AutF0 (P ) is normal in AutF (P ) for every P 6 S0 , since F0 is weakly normal
in F. We need to show that AutF (P )/ AutF0 (P ) is a p-group for every P 6 S0 . Suppose this is
false and let P be a counterexample of maximal order. Our assumption gives P < S0 . Hence,
P < Q := NS0 (P ), and the maximality of P implies that
AutF (Q)/ AutF0 (Q)
is a p-group. Notice that
NAutF (Q) (P )/NAutF0 (Q) (P ) ∼
= NAutF (Q) (P ) AutF0 (Q)/ AutF0 (Q)
6 AutF (Q)/ AutF0 (Q)
and thus NAutF (Q) (P )/NAutF0 (Q) (P ) is a p-group.
If α ∈ HomF (P, S) with α(P ) ∈ F f then conjugation by α induces a group isomorphism
from AutF (P ) to AutF (α(P )). As F0 is weakly normal, we have α(P ) 6 S0 and conjugation
by α takes AutF0 (P ) to AutF0 (α(P )). So upon replacing P by α(P ), we may assume without
loss of generality that P is fully F-normalized. Then P is also fully F0 -normalized by [Asc08,
Lemma 3.4(5)]. By the Sylow axiom, AutS0 (P ) is a Sylow p-subgroup of AutF0 (P ). So the
Frattini argument yields
AutF (P ) = AutF0 (P )NAutF (P ) (AutS0 (P ))
and thus
AutF (P )/ AutF0 (P ) ∼
= NAutF (P ) (AutS0 (P ))/NAutF0 (P ) (AutS0 (P )).
By the extension axiom for F and F0 , each element of NAutF (P ) (AutS0 (P )) extends to an automorphism of AutF (Q), and each element of NAutF0 (P ) (AutS0 (P )) extends to an automorphism of
AutF0 (Q). Therefore, the map
Φ : NAutF (Q) (P ) → NAutF (P ) (AutS0 (P )), ϕ 7→ ϕ|P
is an epimorphism which maps NAutF0 (Q) (P ) onto NAutF0 (P ) (AutS0 (P )). Hence,
AutF (P )/ AutF0 (P ) ∼
= NAutF (P ) (AutS0 (P ))/NAutF0 (P ) (AutS0 (P ))
∼
= NAut (Q) (P )/NAut (Q) (P ) ker(Φ).
F

F0

We have seen above that NAutF (Q) (P )/NAutF0 (Q) (P ) is a p-group, and therefore also
NAutF (Q) (P )/NAutF0 (Q) (P ) ker(Φ)
is a p-group. Hence, AutF (P )/ AutF0 (P ) is a p-group, and this contradicts our assumption that
P is a counterexample.

10

4.1. Extensions to the bottom. A central extension of a fusion system F0 is a fusion system
F such that F/Z ∼
= F0 for some subgroup Z 6 Z(F). The central extension is said to be perfect
if F = O p (F). Linckelmann has shown that the Schur multiplier of a Benson-Solomon system is
trivial.
Theorem 4.2 (Linckelmann). Let F be a perfect central extension of a Benson-Solomon fusion
system F0 . Then F = F0 .
Proof. This follows from Corollary 4.4 of [Lin06a] together with the fact that Spin7 (q) has Schur
muliplier of odd order when q is odd [GLS98, Tables 6.1.2, 6.1.3].

4.2. Extensions to the top. The next theorem describes the possible extensions (S, F) of a
Benson-Solomon system (S0 , F0 ). The particular hypotheses are best stated in terms of the
generalized Fitting subsystem of Aschbacher [Asc11], but they are equivalent to requiring that
F0 E F and CS (F0 ) 6 S0 , where CS (F0 ) is the centralizer constructed in [Asc11, §6]. This latter
formulation is sometimes expressed by saying that F0 is centric normal in F.
k

Theorem 4.3. Let F0 = FSol (52 ) be a Benson-Solomon system over S0 .
(a) If F is a saturated fusion system over S such that F ∗ (F) = F0 , then F0 = O 2 (F), S splits
over S0 , and the map S/S0 → Out(F0 ) induced by conjugation is injective.
(b) Conversely, given a subgroup of A 6 Out(F0 ) ∼
= C2k , there is a saturated fusion system
∗
F over some 2-group S such that F (F) = F0 and the map S/S0 → Out(F0 ) induced by
conjugation on S0 has image A. Moreover, the pair (S, F) with these properties is uniquely
determined up to isomorphism.
If L0 is a centric linking system associated to F0 , then AutL0 (S0 ) = S0 , and the p-group
S can be chosen to be the preimage of A in Aut(L0 ) under the quotient map from Aut(L0 )
to Out(L0 ) ∼
= Out(F0 ).
Proof. Let F be a saturated fusion system over S such that F ∗ (F) = F0 . Set F1 = F0 S, the
internal extension of F0 by S, as in [Hen13] or [Asc11, §8]. According to [AOV12, Proposition 1.31],
there is a normal pair of linking systems L0 E L1 , associated to the normal pair F0 E F1 .
Furthermore, L0 E L1 can be chosen such that L0 is a centric linking system. There is a natural
map from AutL1 (S0 ) to Aut(L0 ) which sends a morphism ϕ ∈ AutL1 (S0 ) to conjugation by ϕ.
(So the restriction of this map to AutL0 (S0 ) is the conjugation map described in §§2.2.2.)
The centralizer CS (F0 ) depends a priori on the fusion system F, but it is shown in [Lyn15,
Lemma 1.13] that it does not actually matter whether we form CS (F0 ) inside of F or inside of F1 .
Moreover, since F ∗ (F) = F0 , it follows from [Asc11, Theorem 6] that CS (F0 ) = Z(F0 ) = 1. Thus,
conj

by a result of Semeraro [Sem15, Theorem A], the conjugation map AutL1 (S0 ) −−→ Aut(L0 ) is
injective. By Lemma 3.4, we have S0 = AutL0 (S0 ) via the inclusion functor δ1 for L1 . By
Theorem 3.10, Aut(L0 ) is a 2-group which splits over S0 . Moreover, by the same theorem, we
have that CAut(L0 ) (S0 ) 6 S0 and Out(L0 ) ∼
= Out(F0 ) is cyclic. Since (δ1 )S0 (S) ∼
= S is a Sylow
2-subgroup of AutL (S0 ) by [Oli10, Proposition 4(d)], we can conclude that
S0 = AutL0 (S0 ) E AutL1 (S0 ) = S,
via the inclusion functor δ1 for L1 . Moreover, it follows that S splits over S0 , and CS (S0 ) 6 S0 .
The latter property means that the map
S/S0 → Out(F0 )
is injective. In particular, S/S0 is cyclic as Out(F0 ) is cyclic.
11

Next, we show that O2 (F) = F0 . Fix a subgroup P 6 S, and let α ∈ AutF (P ) be an
automorphism of odd order. Then α induces an odd-order automorphism of the cyclic 2-group
P/(P ∩ S0 ) ∼
= P S0 /S0 6 S/S0 . This automorphism must be trivial, and so [P, α] 6 S0 . Hence,
2
[P, O (AutF (P ))] 6 S0 for all P 6 S. Since hyp(F0 ) = S0 , we have hyp(F) = S0 . Note that
AutF (S0 ) is a 2-group as AutF (S0 ) 6 Aut(F0 ) and Aut(F0 ) is a 2-group by Theorem 3.10.
Therefore O 2 (F) = F0 by Lemma 4.1. We conclude that F1 = F by the uniqueness statement
in [Hen13, Theorem 1]. This completes the proof of (a). Moreover, we have seen that the following
property holds for any normal pair L0 E L attached to F0 E F:
(4.4)

conj

S0 = AutL0 (S0 ) E AutL1 (S0 ) = S and S −−−→ Aut(L0 ) is injective.

Finally, we prove (b). Fix a centric linking system L0 associated to F0 with inclusion functor
δ0 . Let S 6 Aut(L0 ) be the preimage of A under the quotient map to Out(F0 ). We will identify
S0 with δ0 (S0 ) so that S0 = AutL0 (S0 ) by Lemma 3.4. Write ι : S0 → Aut(L0 ), s 7→ cs for map
sending s ∈ S0 to the automorphism of L0 induced by conjugation with s in L0 . Then ι(S0 ) is
normal in S. Let χ : S → Aut(S0 ) be the map defined by α 7→ ι−1 ◦ cα |ι(S0 ) ◦ ι; i.e. χ corresponds
to conjugation in S if we identify S0 with ι(S0 ). We argue next that the following diagram
commutes:

(4.5)

ι

// Aut(L0 )
✇;;
incl ✇✇✇
α7→αS0
ι
✇
✇✇

 ✇✇✇ χ
// Aut(S0 )
S

S0

The upper triangle clearly commutes. Observe that α◦ι(s)◦α−1 = α◦cs ◦α−1 = cαS0 (s) = ι(αS0 (s))
for every s ∈ S0 and α ∈ S. Hence, for every α ∈ S and s ∈ S0 , we have (ι−1 ◦ cα |ι(S0 ) ◦ ι)(s) =
ι−1 (α ◦ ι(s) ◦ α−1 ) = αS0 (s) and so the lower triangle commutes.
We will now identify S0 with its image in S under ι, so that ι becomes the inclusion map and
χ corresponds to the map S → Aut(S0 ) induced by conjugation in S. As the above diagram
commutes, it follows then that the diagram in [Oli10, Theorem 9] commutes when we take Γ = S
and τ : S → Aut(L0 ) to be the inclusion. Thus, by that theorem, there is a saturated fusion
system F over S in which F0 is weakly normal, and there is a corresponding normal pair of
linking systems L0 E L (in the sense of [AOV12, §1.5]) such that S = AutL (S0 ) has the given
action on L0 (i.e. the automorphism of L0 induced by conjugation with s ∈ S in L equals the
automorphism s of L0 ). By the same theorem, the pair (F, L) is unique up to isomorphism of
fusion systems and linking systems with these properties. Since F0 is simple [Lin06b], F0 is in
fact normal in F by a result of Craven [Cra11, Theorem A]. Thus, since CS (F0 ) 6 CS (S0 ) 6 S0 ,
it is a consequence of [Asc11, (9.1)(2), (9.6)] that F ∗ (F) = F0 .
So it remains only to prove that (S, F) is uniquely determined up to an isomorphism of fusion
systems. Let F ′ be a saturated fusion system over a p-group S ′ such that F ∗ (F ′ ) = F0 , and such
that the map S ′ /S0 → Out(F0 ) induced by conjugation has image A. Then by (a), F0 = Op (F ′ ).
So by [AOV12, Proposition 1.31], there is a normal pair of linking systems L′0 E L′ associated to
the normal pair F0 E F ′ . Moreover, we can choose L′0 to be a centric linking system. Since a
centric linking system attached to F0 is unique, there is an isomorphism θ : L′0 → L0 of linking
systems. We may assume that the set of morphisms which lie in L′ but not in L′0 is disjoint from
the set of morphisms in L0 . Then we can construct a new linking system from L′ by keeping
every morphism of L′ which is not in L′0 and replacing every morphism ψ in L′0 by θ(ψ), and then
carrying over the structure of L′ in the natural way. Thereby we may assume L′0 = L0 . So we are
12

given a normal pair L0 E L′ attached to F0 E F ′ . By (4.4) applied with L′ and F ′ in place of F
and L, we have S0 = AutL0 (S0 ) E AutL′ (S0 ) = S ′ via the inclusion functor δ′ of L′ . Let
τ : S ′ → Aut(L0 )
be the map taking s ∈ S ′ to the automorphism of L0 induced by conjugation with s in L′ . Again
using (4.4), we see that τ is injective. Note also that τ restricts to the identity on S0 if we
identify S0 with ι(S0 ) as above. Recall that the map S ′ /S0 → Out(F0 ) induced by conjugation
has image A. So Theorem 3.10 implies τ (S ′ ) = S, i.e. we can regard τ as an isomorphism
τ : S ′ → S. So replacing (S ′ , F ′ ) by (S, τF ′ ) and then choosing L0 E L′ as before, we may assume
S = S ′ . So F ′ is a fusion system over S with F0 E F ′ , and L0 E L′ is a normal pair of linking
systems associated to F0 E F ′ such that AutL′ (S0 ) = S via δ′ . Let s ∈ S. Recall that τ (s) is
the automorphism of L0 induced by conjugation with s in L′ . Observe that the automorphism of
S0 = AutL0 (S0 ) induced by τ (s) equals just the automorphism of S0 induced by conjugation with
s in S. Similarly, the automorphism s of L0 equals the automorphism of L0 given by conjugation
with s in L, and so induces on S0 = AutL0 (S0 ) just the automorphism given by conjugation with
s in S. Theorem 3.10 gives CAut(L0 ) (S0 ) 6 S0 and this implies that any two automorphisms of
L0 , which induce the same automorphism on S0 , are equal. Hence, τ (s) = s for any s ∈ S. In
other words, S = AutL′ (S0 ) induces by conjugation in L′ the canonical action of S on L0 . The
uniqueness of the pair (F, L) implies now F ′ ∼
= L. This shows that (S, F) is uniquely
= F and L′ ∼
determined up to isomorphism.

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[Asc08]
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[Ben98]

[BLO03]
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[COS08]

[Cra11]
[DW93]
[GL83]
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E-mail address: ellen.henke@abdn.ac.uk
E-mail address: justin.lynd@abdn.ac.uk
Institute of Mathematics, University of Aberdeen, Fraser Noble Building, Aberdeen AB15 5LY,
United Kingdom

14

