Fusion systems realizing certain Todd modules
-
Bob Oliver
Abstract
We study a certain family of simple fusion systems over finite 3-groups, ones that involve Todd modules of the Mathieu groups
Fix a prime 𝑝.
A fusion system over a finite 𝑝-group 𝑆 is a category whose objects are the subgroups of 𝑆, and whose morphisms are injective homomorphisms between the subgroups satisfying certain axioms first formulated by Puig [44], and modeled on the Sylow theorems for finite groups.
The motivating example is the fusion system of a finite group 𝐺 with
The general theme in this paper is to study fusion systems over finite 𝑝-groups 𝑆 that contain an abelian subgroup
This paper is centered around one family of examples: those where
Let ℱ be a saturated fusion system over a finite 3-group 𝑆 with an elementary abelian subgroup
Theorem A is proven below as Theorem 4.14 (case (i)) and Theorem 5.20 (cases (ii) and (iii)).
As one consequence of these results, we give new 3-local characterizations of the three Conway groups as well as of
All three cases of Theorem A have already been shown in earlier papers using very different methods.
In [48, Theorem A], van Beek determined (among other results) all fusion systems ℱ over a Sylow 3-subgroup of
The proof of Theorem A is straightforward, following a program that also seems to work in many other cases.
Set
We then construct a special subgroup
(of order 3 or 9) and
Theorem A involves just one special case of the following general problem.
Given a prime 𝑝, a finite group
We want to know whether a given module can be realized in this sense, and if so, list all of the distinct saturated fusion systems that realize it.
In the papers [32, 17, 35], we studied this question under the additional assumption that
As pointed out by the referee, Theorem A in this paper is closely related to the list of amalgams by Papadopoulos in [38]. It seems quite possible that the results in this paper can be used to strengthen or generalize the main theorem in [38], but if so, that will have to wait for a separate (short) paper.
General definitions and properties involving saturated fusion systems are surveyed in Section 1, while the more technical results needed to carry out the program described above are listed in Section 2.
In Section 3, we set up some notation for working with Todd modules for
Notation and terminology
Most of our notation for working with groups is fairly standard.
When
Other notation used here includes the following:
1 Background
We begin with a survey of the basic definitions and terminology involving fusion systems that will be needed here, such as normalizer fusion systems, the Alperin–Goldschmidt fusion theorem for fusion systems, and the model theorem. Most of these definitions and results are originally due to Puig [44].
1.1 Basic definitions and terminology
A fusion systemℱ over a finite 𝑝-group 𝑆 is a category whose objects are the subgroups of 𝑆, and whose morphism sets
every morphism in ℱ factors as an isomorphism in ℱ followed by an inclusion.
Let ℱ be a fusion system over a finite 𝑝-group 𝑆.
Two subgroups
A subgroup
for each
The fusion system ℱ is saturated if it satisfies the following two conditions.
(Sylow axiom) For each subgroup
(Extension axiom) For each isomorphism
In the following lemma, we describe another important property of fully normalized subgroups.
Lemma 1.2 ([6, Lemma I.2.6 (c)])
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆.
Then, for each
We next recall a few more classes of subgroups in a fusion system.
As usual, for a fixed prime 𝑝, a proper subgroup 𝐻 of a finite group 𝐺 is strongly 𝑝-embedded if
Let ℱ be a fusion system over a finite 𝑝-group 𝑆.
For
𝑃 is ℱ-centric if
𝑃 is ℱ-essential if 𝑃 is ℱ-centric and fully normalized in ℱ, and the group
𝑃 is weakly closed in ℱ if
𝑃 is strongly closed in ℱ if, for each
𝑃 is normal in ℱ (
𝑃 is central in ℱ if each morphism in ℱ extends to a morphism that sends 𝑃 to itself via the identity. Let
Clearly, if 𝑃 is weakly closed in ℱ, then it must be normal in 𝑆.
It follows immediately from the definitions that if
The following notation is useful when referring to some of these classes of subgroups.
For each fusion system ℱ over a finite 𝑝-group 𝑆, define
1.2 The Alperin–Goldschmidt fusion theorem for fusion systems
The following is one version of the Alperin–Goldschmidt fusion theorem for fusion systems. This theorem is our main motivation for defining ℱ-essential subgroups here.
Theorem 1.5 ([6, Theorem I.3.6])
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆.
Then each morphism in ℱ is a composite of restrictions of automorphisms
Equivalently, Theorem 1.5 says that
In fact, up to ℱ-conjugacy, the essential subgroups form the smallest possible set of subgroups that generate ℱ.
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆, and let 𝒯 be a set of subgroups of 𝑆 such that
Proof
Fix
We will prove that
Fix
such that
such that
By Lemma 1.2 and since
where we take
Then
For each 𝑖,
as claimed. ∎
The next two lemmas give different conditions for a subgroup to be normal in a fusion system. Both are consequences of Theorem 1.5.
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆.
A subgroup
Proof
This is essentially the equivalence (a) ⇔ (c) in [6, Proposition I.4.5]. ∎
In general, strongly closed subgroups in a saturated fusion system need not be normal. The next lemma describes one case where this does happen.
Lemma 1.8 ([6, Corollary I.4.7 (a)])
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆.
If
1.3 Normalizer fusion subsystems and models
If ℱ is a fusion system over a finite 𝑝-group 𝑆, then a fusion subsystem
One case where this is straightforward is the construction of normalizers and centralizers of subgroups in a fusion system.
Let ℱ be a fusion system over a finite 𝑝-group 𝑆.
For each
It follows immediately from the definitions that a subgroup
Theorem 1.10 ([6, Theorem I.5.5])
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆, and fix
We next look at models for constrained fusion systems and, in particular, for normalizer fusion subsystems of centric subgroups.
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆.
The fusion system ℱ is constrained if there is a subgroup
A model for a constrained fusion system ℱ over 𝑆 is a finite group 𝑀 with
By the model theorem (see [6, Theorem III.5.10]), every constrained fusion system has a model, unique up to isomorphism. We will need this only in the following situation.
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆.
Then, for each
Furthermore, 𝑀 is unique in the following sense: if
Proof
The subsystem
1.4 Subsystems of index prime to 𝑝
We next turn to fusion subsystems of index prime to 𝑝.
By analogy with groups, this really corresponds to subgroups of a finite group 𝐺 that contain
Let ℱ be a fusion system over a finite 𝑝-group 𝑆.
A fusion subsystem
There is clearly always a smallest fusion subsystem of ℱ of index prime to 𝑝: namely, the subsystem
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆.
Then there is a (unique) smallest saturated fusion subsystem
Proof
See [6, Theorem I.7.7] or [11, Theorem 5.4] for the existence and uniqueness of
In fact, the theorems in [6, 11] cited above both describe the subsystem
For each saturated fusion system ℱ over a finite 𝑝-group 𝑆, we have
Proof
By Theorem 1.14, if
We also need the following result, which gives a more precise description of
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆 such that
no intersection of two distinct members of
for each
Proof
For each
(a) Fix
Let 𝑃 be such that
Conversely, if
(b) Set
as (not necessarily saturated) fusion systems over 𝑆.
Thus
In each case, either 𝑅 is the unique member of
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In particular, this shows that the subgroup
the first equality by [6, Theorem I.7.7], the second since
One can also show that
1.5 Quotient fusion systems
Quotient fusion systems of ℱ over 𝑆 are formed by dividing out by a subgroup of 𝑆, not by a fusion subsystem of ℱ.
Let ℱ be a fusion system, and assume
We refer to [16, Proposition II.5.11] for the proof that
If 𝑄 is strongly closed in ℱ, then every morphism
We will need the following lemma, comparing essential subgroups in ℱ and in
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆, and fix
Proof
If
the natural map
Point (a) is clear since
The natural map
By (a), it suffices to prove (c) when
Conversely, assume
If ℱ is a saturated fusion system over 𝑆 and
In the following lemma, we give two conditions under which
does imply that
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆, and let
Proof
Since
2 General lemmas
As noted in the introduction, in our general setting, we want to analyze a saturated fusion system ℱ over a finite 𝑝-group 𝑆 with an abelian subgroup
In practice, we do not get very far without knowing that the subgroup 𝐴 is normal in 𝑆 and weakly closed in ℱ, and this should perhaps be included in our general assumptions.
But in many cases, it follows easily from the weaker assumptions on 𝐴 and
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆, and let
Proof
Assume otherwise: then
The importance of 𝐴 being weakly closed in our general situation is illustrated by the following lemma.
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆, and assume
If
For each
Hence each
No element of
Proof
(a) Assume
(b) Assume
(c) Assume that
In many of the cases we want to consider, the assumptions we choose on 𝐴 and on
2.1 Proving that
F
=
⟨
N
F
(
A
)
,
C
F
(
Z
)
⟩
When analyzing fusion systems in our setting, we first check whether
The following lemma will be our tool for doing this.
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆, let
Then either
Proof
Set
The second inclusion is clear:
By Lemma 1.2 and as
If
If
If
Assume
For each
Note in particular the following special case of Proposition 2.3.
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆, let
Assume that
Proof
By assumption,
2.2 Normality of subgroups
The results in this subsection will be useful when showing that certain subgroups, especially abelian subgroups, are strongly closed or normal in a fusion system.
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆, and let
Proof
Let 𝒲 be the set of pairs
As 𝑄 is not weakly closed in ℱ, there is
Then choose
Now set
The following is a more technical result that will be needed when proving that
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆, and let
Proof
The fusion systems
since they are centric, so
We now apply Corollary 2.7, with
and
Also,
Set
2.3 Equalities between fusion systems
We finish the section with two sets of conditions for showing that two fusion systems over the same 𝑝-group are equal. Proposition 2.11 will be applied to the fusion systems encountered in Section 4, and Proposition 2.13 to those in Section 5.
Let
Assume also that the homomorphism
induced by restriction is surjective.
Then
Proof
Let
Set
where
Now set
The other criterion we give for two fusion systems to be equal applies only to fusion systems satisfying some very restrictive hypotheses, which are stated separately for easier reference.
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆.
Assume
𝐴 is abelian,
Note that
Let
Then
Proof
If Hypotheses 2.12 hold for
Since
which is impossible since 𝛼 permutes the members of
Set
Fix
this subgroup is normal in Θ since all elements in Θ normalize 𝑅.
By points (i) and (ii) in Hypotheses 2.12 and since 𝐴 and 𝑄 are weakly closed, the conclusions of Lemma 1.16 hold for
for
Again, fix
by Lemma 1.16 (a), so
Together with (2.15), this implies that
For
where the last two groups are equal by assumption.
Then, for
by the Frattini argument and the extension axiom (and since
Set
By (2.17), for each
For
the first equality by (2.17), the second by (2.15), and the last since
by Lemma 1.16 (a). The opposite inclusion is clear, so
since
For
where
So, for each
3 Todd modules in characteristic 3
We describe here the notation we use in Sections 4 and 5 to make computations involving Todd modules: first the Todd module for
3.1 The ternary Golay code and the group
2
M
12
We first set up notation for handling the ternary Golay code 𝒢 and its automorphism group
For a finite set
We also set
be the subgroups of permutation automorphisms and monomial automorphisms, respectively: automorphisms that permute the basis
be the canonical projection that sends a monomial automorphism to the corresponding permutation automorphism; thus
Now set
Thus 𝒯 is a 2-dimensional subspace of
More precisely, each linear automorphism of 𝒯 extends to a unique monomial automorphism of
Set
and set
(the “graph” of 𝜉).
Thus, for each
Finally, for
The
among the
These elements in each of these two sets are independent in
The subspace 𝒢 is the ternary Golay code.
We refer to [23, Lemmas 7.8 and 7.9] for more details and more properties.
Note in particular that we have
We next look at automorphisms of 𝒢.
The following notation is used throughout this section and the next.
Set
For
Fix
Let
Define
and set
By [23, Proposition 7.29],
Note the following relations for
To see the last equality, note that, for
where
In particular, these formulas show that the action of
We have
Proof
By construction, we have
For the maximality of
see [15, p. 235] or [4, p. 8].
Note that if we regard
One easy consequence of Lemma 3.7 is that
To simplify later calculations, we next describe 𝒢 and the action of
For
Let 𝒯 be the tetracode subgroup of (3.2).
Choose a map of sets
by setting
for all
Define
for each
We now check that the maps
The following statements hold.
The map
The map
For each
(3.10)Thus it follows that
Proof
(a) For each
We first check that
We next check that (3.5) holds for the images of the elements in
where the first equality holds since
Thus
Now,
Thus
(b) One easily checks that
Thus
(c) For each
where
this proves (3.10). ∎
To simplify computations still farther, we now describe elements in
Let
be the composites of
For each
As a first, very simple application, we describe the Jordan blocks for actions on 𝑨.
There are exactly two conjugacy classes of elements of order 3 in
Proof
Each element of order 3 in
acts on 𝒢 with three Jordan blocks of lengths
acts with two Jordan blocks of length 3.
Thus these elements are in different classes, and each element of order 3 in
The notation developed in this subsection is summarized in Table 2.
Notation used for certain subgroups of
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3.2 Notation for the Todd modules of
M
11
and
A
6
We next set up notation to work with the Todd modules of the groups
Let
are the images of the six elements of weight 12 in
of Lemma 3.9 (a).
We want to identify
To simplify these descriptions, we identify 𝒯 with
We also write
Assume Notation 3.6 and Table 2, and choose an
Set
Set
Also, set
Finally, define
(Recall that we compose from right to left.)
For
Also, for
For easy reference, we summarize in Table 3 some of the basic properties of groups defined in Notation 3.15.
Assume Notation 3.15.
Then, for
and the groups
In particular,
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Proof
By definition (see Notation 3.6 (d)), each element of 𝑵 normalizes the subspace
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Consider, for example, the case
Since
Hence we have
Recall (Notation 3.6 (d)) that
where
In particular,
It remains only to show that
by definition of
We just saw that
and so the inequalities in (3.17) are all equalities. Hence
and the descriptions of
As seen in Lemma 5.2, there are three different representations that appear under Hypotheses 5.1: one of
Assume Notation 3.15.
We summarize here the notation we use for the
(
in such a way that
(
in such a way that
(
in such a way that
Proof
(b) Define
by setting
This is surjective since
Also,
For
The other formulas follow by similar (but simpler) arguments.
(c) The description of the action of
is defined by
(a) This proof is similar to that of (b), except that
defined by setting
This is easily seen to be surjective.
For
and so
The formulas for
4 The Todd module for
2
M
12
We are now ready to look at fusion systems that involve the Todd module for
Set
The main result in this section is Theorem 4.14, where we show that if
Standard results in the representation theory of
Assume Hypotheses 4.1.
Then
Proof
By [24, § 4 and Table 5], the only 6-dimensional faithful
where
Now,
We next check that, under Hypotheses 4.1, 𝑨 is weakly closed in
Assume that
𝑨 is weakly closed in
𝑺 and 𝑀 both split over 𝑨.
Proof
By Lemma 4.2,
(a) If
(b) Choose
We use throughout this section the notation set up in Section 3.1
for working with the Todd module for
Assume Hypotheses 4.1 and Notation 3.6. Identify
where
be the isomorphism defined by (3.11). Thus
for all
and set
4.1 Some subgroups of
Γ
and 𝑺
We begin by listing the additional notation that will be needed, in particular, notation to describe the subgroups of index 3 in 𝑻.
Define
Define elements
Thus
For
In addition, we set
For
The actions of the
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We have
and
For each
More generally, if
Proof
The descriptions of 𝑍 and
The formulas in point (b) follow easily from those in Table 5.
(Note, for each
If
Note that, for each
but is not extraspecial since
There are exactly three abelian subgroups of 𝑺 of order
If
For each saturated fusion system ℰ over 𝑺 and each
Proof
(a) Assume
and
Thus
and hence
So, by Table 5 again,
In particular, there are at most three subgroups of 𝑺 isomorphic to
(b) Assume that
Conjugation by the element
and
𝑄 is weakly closed only if
If
by Table 5, and so
with equality since
(c) For each
since
by the same lemma.
Since
Point (c) in Lemma 4.7 is not true if one replaces
4.2 Fusion systems involving the Todd module for
2
M
12
We now begin to apply results from Section 2.
Recall that our goal is to describe all fusion systems that satisfy Hypotheses 4.1 with
Assume Hypotheses 4.1 with
Proof
Assume otherwise.
By Proposition 2.3, there are subgroups
Fix
Case 1: Assume first that
Then
Thus
Case 2: Assume that
If
Since
Now assume
Set
Since
Also, we have
Case 3: Finally, assume that
From the formulas in Table 5 again, we see that
We next show that
Assume Hypotheses 4.1, and Notations 4.4 and 4.5, and set
𝑸 is weakly closed in
𝑸 is normal in
𝑍 and
Proof
(c) Since
Recall (Lemma 3.13) that there are two conjugacy classes of elements of order 3 in
As noted earlier (see [24, § 4]), while 𝑨 is not isomorphic
to its dual
so
By the description of the action of 𝑵 on 𝑨 in Notation 4.4, 𝑵 normalizes the subgroup
(d) Since
As noted in the proof of (c), the action of
Then 𝑨 splits as a direct sum of the three irreducible
of which only
(b) Since
(a) We first check that
Assume otherwise: assume
If 𝑸 is not weakly closed, then there are
We next claim that
Fix such a 𝛽.
Since
In particular, we have
Suppose that
We are now ready to prove that
Assume Hypotheses 4.1 and Notation 4.5, and again set
Proof
For
Define inductively
by setting
and
Assume
if
We will show that this is impossible. Fix an element
Thus
To see these, note first that if
Since
Since
In all cases,
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If
and hence
By the above relations, we have
So no
If
Let
In particular,
If
If
and so both of these are characteristic in 𝑅.
Since
If
If
If
In all cases,
and we also get a contradiction in this case.
If
We can now determine
Assume Hypotheses 4.1 and Notation 4.5. Then
Also,
where the action of
Proof
The model 𝑀 for
(Lemmas 4.2 and 4.3 (b)).
Since 𝑸 is weakly closed in
where
is now immediate, where the action of
Since
Since
We are now ready to identify all fusion systems satisfying Hypotheses 4.1.
Let
Assume also that
Proof
By Lemma 4.2,
Set
Let 𝑀 be a model for
Set
Since
by Lemma 4.13.
By Lemma 4.13,
where the action of
So
Thus
Since
The 3-fusion system of
5 Todd modules for
M
10
and
M
11
We now look at Todd modules for the Mathieu groups
Set
We will see in Lemma 5.4 that 𝑨 is weakly closed in
The irreducible
There are exactly one isomorphism class of faithful 4-dimensional
Proof
We refer for simplicity to [27, p. [4]] for the table of characters of
By [26, § 7A], there are exactly two isomorphism classes of irreducible 5-dimensional
Of the two distinct 5-dimensional
As noted in the proof of Lemma 5.2, the 4-dimensional
5.1 Preliminary results
The main goal in this subsection is to show that
Assume Hypotheses 5.1 and Notation 3.15 as well as the notation in Lemma 3.18.
Identify
For later reference, we collect in Table 7 some easy computations involving some of the subgroups of 𝑨 and
In all cases,
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The next lemma gives a first easy consequence of the computations in Table 7.
Assume that
Proof
By Lemma 5.2, 𝑨 is one of the
The following properties will also be needed.
Assume Hypotheses 5.1 and Notation 5.3.
In the
In all cases, if
Proof
(a) By Lemma A.5, for
(b) Assume otherwise: assume
Assume we are in the
Now assume we are in the
Assume
In Notation 5.3, we identified
Assume Hypotheses 5.1 and Notation 5.3.
Then, for
Proof
For
where the second equality holds by Lemma 3.7.
The maximality of
Now let
These inequalities are equalities by Table 3 and since
so
We can now begin to apply some of the lemmas in Section 2.
Proof
Assume otherwise, and recall that
Set
Fix
Case 1: Assume first that
If we are in the
In the
Case 2: Now assume that
Assume first we are in the
So there are
Now assume we are in the
5.2 The subgroup
Q
⊴
C
F
(
Z
)
So far, we have shown that
Assume Hypotheses 5.1 where
Proof
Assume Notation 5.3. Define
Recall that
We will prove that
Since
It thus remains to prove (5.9).
Set
Let
Since 𝑸 is the only member of 𝒬, it is normalized by
normalizes
In the
by Lemma 1.18, and hence
As
Assume 𝑅 is not of exponent 3, and set
It remains to show that
(Note that
and
for each
Hence
We list some of the properties of these subgroups
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One easy consequence of Proposition 5.8 is that
Assume Hypotheses 5.1 where
Proof
Let
If
Since 𝑺 splits over 𝑨, it follows from Gaschütz’s theorem (see [3, (10.4)]) that 𝑀 also splits over 𝑨. ∎
Recall that, for
Assume Hypotheses 5.1 and Notations 3.15 and 5.3.
Let 𝑀 be a model for
where
One easily sees that 𝑸 is special with
Recall Notation 3.15 and Lemma 3.16:
Assume Hypotheses 5.1 and Notation 5.11, and also that
Proof
Since 𝑀 is a model for
Each pair of distinct Sylow 3-subgroups of
By Proposition 5.7,
Recall that
By Lemma 5.2, 𝑨 is absolutely irreducible as an
as earlier.
Thus the centralizer in
is also a 2-group, and so Hypothesis 2.12 (iii) holds. ∎
The following notation for elements in 𝑸 will be useful.
For
Thus each element of 𝑸 is represented by a unique triple
We list in Table 9 some of the relations among such triples: all of these are immediate consequences of the definition in Notation 5.13 and the relations in Lemma 3.18.
Here,
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The next two lemmas give more information about
Assume Hypotheses 5.1, and Notations 5.3 and 5.11, with
Proof
Since
by Lemma 3.16, and 𝛽 acts on 𝑺 via
For calculations in
for
We identify
and hence
Here, each element of the form
It remains to prove the lemma for automorphisms of the form
Each of the
on
by (5.17).
Since
Recall that
The next lemma is needed to handle the cases where
Assume Hypotheses 5.1 and Notation 5.11, where
If we are in the
If, furthermore,
If we are in the
Proof
Recall that
(a) Assume we are in the
acts on
Now let
Now fix subgroups
and an element
(b) Assume we are in the
where
and hence this group has two subgroups of order 8, generated by
By comparing characteristic polynomials or traces for the actions of the
Set
by Lemma A.4.
In particular, there is an element
Now let
From the list in [20] of subgroups of
5.3 Fusion systems involving the Todd modules for
M
10
and
M
11
We are now ready to state and prove our main theorem on fusion systems satisfying Hypotheses 5.1.
Let
(Note that (a), (b), and (c) correspond to the
Proof
By Lemma 5.4,
By Proposition 5.8, there is a unique special subgroup
Also,
for short.
If
If
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Thus, in all cases,
In all of the other cases, we refer to the tables in [4, pp. 7–40], which show that
Let 𝑀 be a model for
Choose an isomorphism
such that
The fusion system
By Lemma 5.12, the fusion systems
and note that
The
By Lemma 5.14,
The
by that lemma.
Set
(see Lemma 3.16), and set
The
The automizers of the subgroups 𝑨 and 𝑸 in each case of Theorem 5.20 are described more explicitly in Table 11.
We refer again to [4, pp. 7–40] in all cases except that of
In all cases,
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Note that, by [12, Theorem A (a), (d)], the 3-fusion system of
6 Some 3-local characterizations of the Conway groups
We finish with some new 3-local characterizations of the three Conway groups,
We first combine Theorem 4.14 with the 3-local characterization of
Let 𝐺 be a finite group.
Assume
𝐴 is not strongly closed in 𝑆 with respect to 𝐺; and
Proof
By Salarian’s theorem [45, Theorem 1.1], to show that
Set
Condition (ii) holds by (1). Also,
where
Set
by Lemma 4.12.
The action of
Now,
The following 3-local characterization of
Let 𝐺 be a finite group.
Assume
𝐴 is not strongly closed in 𝑆 with respect to 𝐺; and
Proof
By the theorem of Korchagina, Parker, and Rowley [28, Theorem 1.1], to show that
Set
Condition (ii) holds by (1), and since
Set
Now, 𝑄 is ℱ-centric by Proposition 5.8, so
Finally, we combine Theorem 5.20 with results of Parker, Rowley, and Stroth, to get some new 3-local characterizations of
Let 𝐺 be a finite group, fix
𝐴 is not strongly closed in 𝑆 with respect to 𝐺; and
If
If
Proof
Set
Set
But if
Set
Now, 𝑄 is ℱ-centric by Proposition 5.8, so
If
If
𝑍 is not weakly closed in 𝑆 with respect to 𝐺.
If
So
Funding source: Centre National de la Recherche Scientifique
Award Identifier / Grant number: UMR 7539
Funding source: Isaac Newton Institute for Mathematical Sciences
Award Identifier / Grant number: GRA2
Funding source: Engineering and Physical Sciences Research Council
Award Identifier / Grant number: EP/K032208/1
Funding statement: B. Oliver is partially supported by UMR 7539 of the CNRS. Part of this work was carried out at the Isaac Newton Institute for Mathematical Sciences during the programme GRA2, supported by EPSRC grant nr. EP/K032208/1.
A Some special 𝑝-groups
In this section, we give a few elementary results on special or extraspecial 𝑝-groups and their automorphism groups.
Most of them involve 𝑝-groups of the form
Fix a prime 𝑝, and let 𝑄 be a finite nonabelian 𝑝-group such that
The quotient group
If 𝑝 is odd, then there is a homomorphism
Assume
Assume
Proof
Set
(a) For each
(b) For each
(Recall that
(c) Assume
Fix
(d) Assume
Now assume
In the rest of the section, we prove some more specialized results on certain special 𝑝-groups.
Recall that, for each prime power 𝑞 and each
Let 𝑝 be an odd prime, and set
be the natural homomorphism.
We regard
The image
We have
Let
More precisely, each
Proof
(a), (b) See [39, Proposition 5.3].
(c) For short, set
Fix
Thus
Fix
A similar argument shows that each
When working with automorphisms of extraspecial groups
Let 𝑉 be a 4-dimensional
The elements
The elements
Proof
The conjugacy classes of elements of order 3 in
Fix
Fix a basis
If 𝑔 and ℎ have matrices
There are 40 maximal isotropic subspaces, each of which is fixed by three subgroups of the form
Finally, we consider certain subgroups of extraspecial groups of order
Assume 𝑄 is extraspecial of order
be such that
𝑃 is isomorphic to
Proof
Set
If 𝑅 is in one of the two classes of maximal parabolic subgroups, then
Since
If
If
Assume
Since
The groups
Thus 𝑃 is isomorphic to
We finish the section with the following well-known and elementary lemma.
Fix a prime 𝑝.
Let 𝐺 be a finite 𝑝-group, let
Proof
Set
(in additive notation).
So
B Strongly 𝑝-embedded subgroups
We collect here some of the basic properties, especially for odd primes 𝑝, of finite groups with strongly 𝑝-embedded subgroups. All of the results here are proven independently of the classification of finite simple groups (but see remarks in the proof of Proposition B.10).
Let 𝐺 be a finite group, and let
Proof
Recall (see [25, Theorem X.4.11 (b)]) that 𝐺 has a strongly 𝑝-embedded subgroup if and only if there is a partition
Let 𝐺 be a finite group with a strongly 𝑝-embedded subgroup
Each proper subgroup
For each normal subgroup
Proof
(a) Assume
so
(b) If
The next few lemmas provides different ways of showing that certain groups do not have strongly 𝑝-embedded subgroups.
Fix a finite group 𝐺 containing a strongly 𝑝-embedded subgroup.
Let
In all cases,
If
Proof
Fix
(a) We show this by induction on
(b) Assume
Assume neither
and
since
Thus
The next lemma is an easy consequence of the well-known list of subgroups of
Fix a prime 𝑝 and
Proof
Since
In the next few lemmas,
Let 𝑃 be a finite 𝑝-group, and let
Proof
For each such 𝛼,
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆, and assume
If
If
Proof
Fix
induced by restriction to
Assume that
Recall that
Without loss of generality, in both points (a) and (b), we can assume that the filtration by the
(a) If
contains a strongly 𝑝-embedded subgroup.
(b) Now assume that
The next lemma provides another way to show that certain subgroups of a 𝑝-group 𝑆 cannot be essential in any fusion system over 𝑆.
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆.
Assume
Then
Proof
Assume otherwise: assume 𝑃 is ℱ-essential.
Set
Thus
Since 𝐻 is strongly 𝑝-embedded and
By assumption, we have that
The next lemma gives yet another simple criterion for a subgroup not to be essential.
Again,
Let ℱ be a saturated fusion system over a finite 𝑝-group 𝑆, and fix
Proof
By Lemma B.5
and since
We finish by listing the subgroups of
Fix an odd prime 𝑝, let 𝑉 be a 4-dimensional vector space over
Proof
By Aschbacher’s theorem [2], applied to the finite simple classical group
By Lemma B.1,
Case 1: Assume 𝐺 is contained in a member of Aschbacher’s class
If
If
If
The class
Assume
The symplectic group
Case 2: It remains to check the cases where the image in
The tables in [10] were made using the classification of finite simple groups.
But lists of maximal subgroups of
Elements of order 𝑝: The description of the Jordan blocks for the natural action of
The isomorphism extends to an isomorphism
between automorphism groups, so all elements of order 𝑝 in
Acknowledgements
The author would especially like to thank the referee for the many helpful suggestions, including some involving potential connections with other papers. He would also like to thank the Isaac Newton Institute for its hospitality while the paper was being revised.
-
Communicated by: Christopher W. Parker
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© 2022 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Fusion systems realizing certain Todd modules
- Diagonal embeddings of finite alternating groups
- Vertex-transitive graphs with local action the symmetric group on ordered pairs
- On the Schur multiplier of finite 𝑝-groups of maximal class
- A classification of skew morphisms of dihedral groups
- On closed subgroups of precompact groups
- Inertia of retracts in Demushkin groups
- On arithmetic properties of solvable Baumslag–Solitar groups
Artikel in diesem Heft
- Frontmatter
- Fusion systems realizing certain Todd modules
- Diagonal embeddings of finite alternating groups
- Vertex-transitive graphs with local action the symmetric group on ordered pairs
- On the Schur multiplier of finite 𝑝-groups of maximal class
- A classification of skew morphisms of dihedral groups
- On closed subgroups of precompact groups
- Inertia of retracts in Demushkin groups
- On arithmetic properties of solvable Baumslag–Solitar groups
