RANK 2AMALGAMS AND FUSION SYSTEMS MARTIN VAN BEEK Abstract. We classify fusion systems Fin which OpF 1 and there are

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RANK 2AMALGAMS AND FUSION SYSTEMS

MARTIN VAN BEEK

Abstract. We classify fusion systems Fin which Op(F) = {1}, and there are

two AutF(S)-invariant essential subgroups whose normalizer systems generate

F. We employ the amalgam method and, as a bonus, obtain p-local character-

izations of certain rank 2 group amalgams whose parabolic subgroups involve

strongly p-embedded subgroups.

1. Introduction

For a ﬁnite group Gand a prime pdividing the order of G, the p-fusion category of

Gprovides a means to concisely express properties of the conjugacy of p-elements

within G. Fusion systems may then be viewed as an abstraction of fusion categories

without the need to specify a group G, instead focusing only on the properties of

a particular p-group.

The purpose of this paper is to classify certain fusion systems which are gener-

ated by automorphisms of two subgroups which satisfy certain properties. This is

achieved by identifying a rank two amalgam within the fusion system, and then

utilizing the amalgam method. In this way, the work in this paper may be viewed

not only as a result about fusion systems, but as a result about (not necessarily

ﬁnite) groups (see Theorem C). Furthermore, as an application, the work here will

aid in classifying saturated fusion systems over maximal unipotent subgroups of

ﬁnite groups of Lie type of rank 2. Indeed, this result has already been used to

give a complete description of all saturated fusion systems supported on p-groups

isomorphic to a Sylow p-subgroup of G2(pn) or PSU4(pn), see [vBe21].

The methodology for proving this result breaks down as follows. Firstly, we identify

the automizers of the two distinguished subgroups in our fusion system F. Here

the subgroups in question are essential in Fand, using fusion techniques and

classiﬁcation results concerning groups with strongly p-embedded subgroups, we

can almost completely describe the outer automorphisms induced by F. Then,

using the model theorem, we are able to able to investigate ﬁnite groups whose

fusion categories are isomorphic to normalizer subsystems of the two distinguished

This work formed the majority of author’s PhD thesis at the University of Birmingham under

the supervision of Prof. Chris Parker. The author gratefully acknowledges the support received

from the EPSRC (EP/N509590/1) during this period.

arXiv:2210.01013v1 [math.GR] 3 Oct 2022

2 MARTIN VAN BEEK

essential subgroups. With these two groups in hand, we can then form an amalgam

whose faithful completion realizes F.

This leads to the second part of the analysis. Here, we employ the amalgam

method, building on work began by Goldschmidt [Gol80]. In our interpretation,

we closely follow the techniques developed and reﬁned by Delgado and Stellmacher

[DS85] and a large number of the amalgams we investigate are fortunately already

classiﬁed there. Indeed, several of the amalgams we investigate are unique up to

isomorphism and, as it turns out, this is enough to determine the fusion system up

to isomorphism. However, in some cases, we do not go so far and instead aim only

to bound the order of the p-group on which Fis supported and apply a package

in MAGMA [PS21] which identiﬁes the fusion system. In fact, in two instances

there are no ﬁnite groups which realize the amalgam appropriately and we uncover

two exotic fusion systems, one of which was known about previously by work of

Parker and Semeraro [PS18], and another which has been described in [vBe22].

With that said, given the information we gather about the amalgams, it does not

seem such a stretch to at least provide a characterization of these amalgams up to

some weaker notion of isomorphism.

Within this work, we very often use a K-group hypothesis when investigating

automizers of essential subgroups and a CK-system hypothesis on the fusion system

F. Recall that K-group is a ﬁnite group in which every simple section is isomorphic

to a known ﬁnite simple group. A CK-system is then a saturated fusion system

in which the induced automorphism groups on all p-subgroups are K-groups. At

some stage in the analysis, unfortunately, we make explicit use of the classiﬁcation

of ﬁnite simple groups (CFSG), speciﬁcally when Fis exotic. However, up to

that point, we are still able to determine the size of the p-group on which F

is supported, as well the local actions, within a CK-system hypothesis and only

appeal to the classiﬁcation to prove that the fusion system is exotic. Thus, we

believe this result would still be suitable for use in any investigation of fusion

systems in which induction via a minimal counterexample is utilized. The main

theorem is as follows:

Main Theorem. Let Fbe a local CK-system on a p-group S. Assume that F

has two AutF(S)-invariant essential subgroups E1, E2ESsuch that for F0:=

hNF(E1), NF(E2)iSthe following conditions hold:

•Op(F0) = {1};

•for Gi:= OutF(Ei), if Gi/O30(Gi)∼

=Ree(3),Giis p-solvable or Tis

generalized quaternion, then NGi(T)is strongly p-embedded in Gifor T∈

Sylp(Gi).

Then F0is saturated and one of the following holds:

(i) F0=FS(G), where F∗(G)is isomorphic to a rank 2simple group of Lie

type in characteristic p;

RANK 2 AMALGAMS AND FUSION SYSTEMS 3

(ii) F0=FS(G), where G∼

=M12,Aut(M12),J2,Aut(J2),G2(3) or PSp6(3) and

p= 2;

(iii) F0=FS(G), where G∼

=Co2,Co3,McL,Aut(McL),Suz,Aut(Suz) or Ly

and p= 3;

(iv) F0=FS(G), where G∼

=PSU5(2),Aut(PSU5(2)),Ω+

8(2),O+

8(2),Ω−

10(2),

Sp10(2),PSU6(2) or PSU6(2).2and p= 3;

(v) F0is simple fusion system on a Sylow 3-subgroup of F3and, assuming

CFSG,F0is an exotic fusion system uniquely determined up to isomor-

phism;

(vi) F0=FS(G), where G∼

=Ly,HN,Aut(HN) or Band p= 5; or

(vii) F0is a simple fusion system on a Sylow 7-subgroup of G2(7) and, assuming

CFSG,F0is an exotic fusion system uniquely determined up to isomor-

phism.

We include G2(2)0∼

=PSU3(3), Sp4(2)0∼

=Alt(6) and the Tits groups 2F4(2)0as

groups of Lie type in characteristic 2.

Some explanation is required for the exceptional condition when

OutF(Ei)/O30(OutF(Ei)) ∼

=Ree(3), OutF(Ei) is p-solvable or OutS(Ei) is

generalized quaternion in the above theorem. This assumption ensures that

the centralizers of non-central chief factors in a model of NF(Ei) are p-closed.

This is key in our methodology and facilitates the use of various techniques,

especially coprime action arguments, which are used to deduce OutF(Ei) and

its action on Ei. In the case where OutF(Ei)/O50(OutF(Ei)) ∼

=Sz(32) : 5, this

could also pose a problem. However, in this situation we are able to apply a

transfer argument to dispel this case (see Proposition 5.9). It is easily seen

using the Alperin–Goldschimidt theorem that if Eiis not contained in any other

essential subgroup of F(which we term maximally essential) then the exceptional

condition holds. We anticipate that in in the majority of the applications of the

Main Theorem, this hypothesis can be forced.

In support of the Main Theorem, applying various results from [Cle07], [AOV17],

[HS19], [PS21], [vBe21] and [vBe22], we can also describe Fup to isomorphism

in most cases. It remains to classify the fusion systems supported on a Sylow

p-subgroup of 2F4(2n), 3D4(pn) or PSU5(pn). The important cases to consider are

where the set of essential subgroups is not contained in the set of unipotent radi-

cals of the two maximal parabolic subgroups of the 2F4(2n), 3D4(pn) or PSU5(pn)

arranged to contain S. We expect that this never happens in these examples and,

in the language above, that F=F0.

In the classiﬁcation in Main Theorem, where F0is realizable by ﬁnite group, we

provide only one example of a group which realizes the fusion system. In several

instances, this example is not unique, even amongst ﬁnite simple groups. In par-

ticular, if F0is realized by a simple group of Lie type in characteristic coprime to

p, then there are lots of examples which realize the fusion system, see for instance

4 MARTIN VAN BEEK

[BMO12]. Note also that we manage to capture a large number of fusion systems

at odd primes associated to sporadic simple groups. Indeed, as can be witnessed in

the tables provided in [AH12], almost all of the p-fusion categories of the sporadic

simple groups at odd primes are either constrained, supported on an extraspecial

group of exponent pand so are classiﬁed in [RV04], or satisfy the hypothesis of

the Main Theorem.

It is surprising that in the conclusion of the Main Theorem there are so few exotic

fusion systems. It has seemed that, at least for odd primes, exotic fusion systems

were reasonably abundant. Perhaps an explanation for the apparent lack of exotic

fusion system is that the setup from the Main Theorem somehow reﬂects some

of the geometry present in rank 2 groups of Lie type. Additionally, we remark

that in the two exotic examples in the classiﬁcation, the fusion systems are ob-

tained by “pruning” a particular class of essential subgroups, as deﬁned in [PS21].

Indeed, these essential subgroups, along with their automizers, seem to resemble

Aschbacher blocks, the minimal counterexamples to the Local C(G, T )-theorem

[BHS06]. Most of the exotic fusion systems the author is aware of either have a

set of essentials resembling blocks, or are obtained by pruning a class of essentials

resembling blocks out of the fusion category of some ﬁnite group. For instance,

pearls in fusion systems, investigated in [Gra18] and [GP22], are the smallest ex-

amples of blocks in fusion systems.

The work we undertake in the proof of the Main Theorem may be regarded as a

generalization of some of the results in [AOV13], where only certain conﬁgurations

at the prime 2 are considered. There, the authors exhibit a situation in which a

pair of subgroups of the automizers of pairs of essential subgroup generate a sub-

system, and then describe the possible actions present in the subsystem, utilizing

Goldschmidt’s pioneering results in the amalgam method. With this in mind, we

provide the following corollary along the same lines which, at least with regards

to essential subgroups, may also be considered as the minimal situation in which

a saturated fusion system satisﬁes Op(F) = {1}.

Corollary A. Suppose that Fis a saturated fusion system on a p-group Ssuch

that Op(F) = {1}. Assume that Fhas exactly two essential subgroups E1and

E2. Then NS(E1) = NS(E2)and writing F0:= hNF(E1), NF(E2)iNS(E1),F0is a

saturated normal subsystem of Fand either

(i) F=F0is determined by the Main Theorem;

(ii) pis arbitrary, F0is isomorphic to the p-fusion category of H, where

F∗(H)∼

=PSL3(pn), and Fis isomorphic to the p-fusion category of G

where Gis the extension of Hby a graph or graph-ﬁeld automorphism;

(iii) p= 2,F0is isomorphic to the 2-fusion category of H, where F∗(H)∼

PSp4(2n), and Fis isomorphic to the 2-fusion category of Gwhere Gis

the extension of Hby a graph or graph-ﬁeld automorphism; or

RANK 2 AMALGAMS AND FUSION SYSTEMS 5

(iv) p= 3,F0is isomorphic to the 3-fusion category of H, where F∗(H)∼

G2(3n), and Fis isomorphic to the 3-fusion category of Gwhere Gis the

extension of Hby a graph or graph-ﬁeld automorphism.

As intimated earlier in this introduction, we utilize the amalgam method to classify

the fusion systems in the statement of the Main Theorem. Here, we work in a

purely group theoretic setting and so, as a consequence of the work in this paper,

we obtain some generic results concerning amalgams of ﬁnite groups which apply

outside of fusion systems. We operate under the following hypothesis, and note

that the relevant deﬁnitions are provided in Section 4:

Hypothesis B. A:= A(G1, G2, G12) is a characteristic pamalgam of rank 2

satisfying the following:

(i) for S∈Sylp(G12), NG1(S) = NG2(S)≤G12; and

(ii) writing Gi:= Gi/Op(Gi), Gicontains a strongly p-embedded subgroup and

if Gi/O30(Gi)∼

=Ree(3), Giis p-solvable or Sis generalized quaternion, then

NGi(S) is strongly p-embedded in Gifor S∈Sylp(G12).

It transpires that all the amalgams satisfying Hypothesis B are either weak BN-

pairs of rank 2; or p67, |S|629when p= 2, and |S|6p7when pis odd.

Moreover, in the latter exceptional cases we can generally describe, at least up to

isomorphism, the parabolic subgroups of amalgam. It is worth mentioning that

condition (ii) holds whenever Gihas a strongly p-embedded subgroup and Op0(Gi)

is p-minimal.

What is remarkable about these results is that amalgams produced have “critical

distance” (deﬁned in Notation 5.4) bounded above by 5 . In the cases where the

amalgam is not a weak BN-pair of rank 2, the critical distance is bounded above by

2, and when this distance is equal to 2, the amalgam is symplectic and was already

known about by work of Parker and Rowley [PR12]. We present an undetailed

version of the theorem summarizing the amalgam theoretic results below.

Theorem C. Suppose that A:= A(G1, G2, G12)satisﬁes Hypothesis B. Then one

of the following occurs:

(i) Ais a weak BN-pair of rank 2;

(ii) p= 2,Ais a symplectic amalgam, G1/O2(G1)∼

=Sym(3),G2/O2(G2)∼

(3 ×3) : 2 and |S|= 26;

(iii) p= 2,Ω(Z(S)) EG2,h(Ω(Z(S))G1)G2)i 6≤ O2(G1),O20(G1)/O2(G1)∼

SU3(2)0,O20(G2)/O2(G2)∼

=Alt(5) and |S|= 29;

(iv) p= 3,Ω(Z(S)) EG2,h(Ω(Z(S))G1)i 6≤ O3(G2),O3(G1) = h(Ω(Z(S))G1)i

is cubic 2F-module for O30(G1/O3(G1)) and |S|637; or

(v) p= 5 or 7,Ais a symplectic amalgam and |S|=p6.

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RANK2AMALGAMSANDFUSIONSYSTEMSMARTINVANBEEKAbstract.WeclassifyfusionsystemsFinwhichOp(F)=f1g,andtherearetwoAutF(S)-invariantessentialsubgroupswhosenormalizersystemsgenerateF.Weemploytheamalgammethodand,asabonus,obtainp-localcharacter-izationsofcertainrank2groupamalgamswhoseparabolicsubgroupsinvolvest...

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RANK 2AMALGAMS AND FUSION SYSTEMS MARTIN VAN BEEK Abstract. We classify fusion systems Fin which OpF 1 and there are

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