FUSION SYSTEMS ON SMALL p-GROUPS

DAVID A. CRAVEN AND ADAM GLESSER

Abstract. In this article we study several classes of ‘small’ 2-groups: we complete the classification, started in [26], of all saturated fusion systems on metacyclic p-groups for all primes p. We consider Suzuki 2-groups, and classify all center-free saturated fusion systems on 2-groups of 2-rank 2. We end by classifying all possible F-centric, F-radical subgroups in saturated fusion systems on 2-groups of 2-rank 2.

1. Introduction

In the classification of finite simple groups, a broad division is made between simple groups that are ‘small’ in some sense and those that are ‘large’; one measure of size, less crude than the order, is the p-rank: the rank of a largest elementary abelian subgroup of a Sylow p-subgroup of a group. In this article we will examine the possible fusion on ‘small’ p-groups.

The modern way to describe fusion in finite groups is via saturated fusion systems (see, for example, [5] for background). Remarkably, there are saturated fusion systems that do not arise from finite groups and these exotic systems are an interesting and rare phenomenon, particularly for the prime 2.

One way of hunting for exotic fusion systems is to fix a finite p-group and ask for all possible saturated fusion systems on that p-group. If you can show that one of these does not come from a finite group, you’ve found an exotic system. Our work begins by completing the classification of saturated fusion systems on metacyclic pgroups, started by Stancu in [26] with his description of saturated fusion systems on metacyclic p-groups for odd primes p. If G is a finite group with Sylow p-subgroup P, then we denote the fusion system of G by F_{P} (G).

Theorem 1.1. Let P be a metacyclic 2-group.

(1) If P is dihedral or generalized quaternion, then there are three saturated fusion systems on P.

(2) If P is semidihedral then there are four saturated fusion systems on P.

(3) If P is the abelian group C_{2^{n}} × C_{2^{n}} then there are two saturated fusion systems on P.

(4) If P is none of the above groups, then there is a unique saturated fusion system on P, namely the fusion system F_{P} (P).

In Section 5 we construct all of these saturated fusion systems as fusion systems of finite groups. For odd primes p, any saturated fusion system of a metacyclic p-group P is the fusion system of a group of the form P ⋊ A, where A is a p′-group of automorphisms of P. If every saturated fusion system on a p-group P is of this form, then P is called resistant. We provide a new proof of Stancu’s result that metacyclic p-groups, for p odd, are resistant.

Date: July 9, 2010.

2000 Mathematics Subject Classification. Primary 20D20; Secondary 20D45.

After a short section proving that Suzuki 2-groups – nonabelian 2-groups with an automorphism that permutes the involutions transitively – are resistant, we turn to fusion systems on groups of 2-rank 2. It is shown in [4] that, up to isomorphism, one may reconstruct F from F/Z(F) and P, so it makes sense to study center-free fusion systems. In Section 5, we prove the following result, adapting the fusiontheoretic argument of Alperin in [2].

Theorem 1.2. Let F be a saturated fusion system on a finite 2-group P and suppose that P has 2-rank 2. If Z(F) = 1, then P is dihedral, semidihedral, wreathed, C_{2^{n}} × C_{2^{n}} or the Sylow 2-subgroup of PSU_{3}(4).

This list is very similar to that of the Sylow 2-subgroups of the finite simple groups of 2-rank 2, although we get more abelian 2-groups.

Having determined the center-free fusion systems on 2-groups of 2-rank 2, we consider all saturated fusion systems. In [8], Díaz, Ruiz and Viruel gave a description of all saturated fusion systems on p-groups of p-rank 2 for p odd. The case of 2-rank 2 is not attempted in [8] and there are special difficulties in this case arising from the fact that there are many 2-groups of 2-rank 2.

Here we will start the classification of such saturated fusion systems by exhibiting a classification of all possible F-centric, F-radical subgroups of such a fusion system, that is, classifying all 2-groups of 2-rank 2 that possess a nontrivial automorphism of odd order (the automorphism groups of such groups, essentially, determine the fusion system). This does not appear to have been carried out in the literature, although such situations have been studied in, for example, [27] and [28].

In Section 6 we define 2-groups X_{n} and Y_{n}, each of order 2^{n} for n ≥ 6, which possess three involutions and an outer automorphism of order 3. The next theorem comes from that section. (Our notation for various p-groups is defined at the end of the introduction.)

Theorem 1.3. Let P be a finite 2-group of 2-rank 2. If Aut(P) is not a 2-group, then P is one of the following groups:

(1) C_{2^{n}} × C_{2^{n}} or Q_{8} × C_{2^{n}} for some n ≥ 1;

(2) Q_{8} ∗ C_{2^{n}} (the central product of Q_{8} and C_{2^{n}}) for some n ≥ 2;

(3) Q_{8} ∗ D_{2^{n}} or Q_{8} × Q_{2^{n}} for some n ≥ 3;

(4) X_{n} or Y_{n} for n ≥ 6 (noting that X_{6} ≅ Y_{6});

(5) Q_{8} ≀ C_{2};

(6) the Sylow 2-subgroup of PSU_{3}(4).

For each of these it is possible to analyze the possible fusion systems containing it. If F is a saturated fusion system on a finite 2-group P, and if U denotes an F-essential subgroup of P, then by necessity U is one of the 2-groups above (or possibly Q_{8}); if U is not normal in P then the structure of N_{P} (U) and N_{F}(U) is very restricted. This mirrors the classification in [8], where there are many groups for which an essential subgroup has index p, but very few with larger index.

The organization of this paper is as follows: in the next section are the preliminary results that we need, with the following section proving Theorem 1.1, and considering metacyclic p-groups for p odd. In Section 4 we introduce Suzuki 2groups and prove that they are resistant, and in Section 5 we prove Theorem 1.2. The final section proves Theorem 1.3.

Our notation and definitions are largely standard now, and are taken, for example, from [7]. Note that, as in [7], our maps will be written on the right, and composition is left-to-right. The cyclic group of order q is denoted by C_{q}, the dihedral, semidihedral, and generalized quaternion groups of order 2^{n} are denoted by D_{2^{n}}, SD_{2^{n}} and Q_{2^{n}} respectively, and the Klein 4-group C_{2} × C_{2} is denoted by V_{4}. The wreathed 2-group is defined to be C_{2^{n}} ≀ C_{2}.

2. Preliminaries

We begin by finding sufficient conditions for the maximal subgroups of a finite pgroup P to be isomorphic via automorphisms of P. This will serve two important roles in the paper. First, for many of the groups in this paper, the structure of the maximal subgroups is understood, allowing us to verify immediately whether the sufficient conditions hold. Second, in this situation, any proper characteristic subgroup of P is contained in Φ(P), the Frattini subgroup of P. This, for instance, allows us to use the famous result of Burnside (see [10, Theorem 5.1.14]) that the kernel of the natural map from Aut(P) to Aut(P/Φ(P)) is a p-group; this same result holds, of course, for any characteristic subgroup of P contained in Φ(P).

Proposition 2.1. Let P be a 2-generator finite p-group and let A be a subgroup of Out(P). If |A| is divisible by p and O_{p}(A) = 1, then the preimage of A in Aut(P) acts transitively on the set of maximal subgroups of P.

Proof. Let Ã denote the full preimage of A under the map Aut(P) → Out(P), and notice that Inn(P) is the kernel of the map Ã → Aut(P/Φ(P)) ≅ GL_{2}(p). Thus A is isomorphic to a subgroup of G = GL_{2}(p). We claim that (identifying A with its image in G) A contains SL_{2}(p). Since O_{p}(A) = 1, the number of Sylow p-subgroups of A must be p + 1 (as there are that many in G), and so |A| ≥ p(p + 1); from the structure of GL_{2}(p), we see that the same holds for A ∩ SL_{2}(p). In particular, the index of A ∩ SL_{2}(p) in SL_{2}(p) is at most p − 1, which immediately leads to a contradiction if p ≥ 5 as SL_{2}(p) is quasisimple and p | |SL_{2}(p)|. If p = 2, then A ∩ SL_{2}(p) = S_{3}, and it is easy to see that SL_{2}(3) possesses no subgroup of index 2. Therefore A ∩ SL_{2}(p) = SL_{2}(p).

Since SL_{2}(p) acts transitively on the projective line, we see that A permutes transitively the p + 1 different 1-dimensional subspaces of the 2-dimensional 𝔽_{p}-vector space P/Φ(P); therefore Ã must permute the maximal subgroups of P transitively. In particular, the maximal subgroups are isomorphic. □

The conditions of this proposition were set up in such a way that if Q is an F-essential subgroup of a saturated fusion system F, and Q is 2-generator, then all maximal subgroups of Q are isomorphic. Notice that all we really needed in the proof was that A contained elements of order p + 1 in SL_{2}(p), since they permute the 1-dimensional subspaces transitively. In the case where p = 2, we can therefore say something stronger.

Corollary 2.2. Let P be a 2-generator 2-group. If Aut(P) is not a 2-group, then Aut(P) acts transitively on the maximal subgroups of P.

Proof. Since GL_{2}(2) ≅ S_{3}, Out(P) contains an element of order p + 1 = 3 which, by the above remark, transitively permutes the three subspaces of dimension 1. □

In the next corollary, we collect several consequences of Proposition 2.1 and Corollary 2.2. Recall from [10, Theorem 5.4.5] that the 2-groups with maximal class are the dihedral, semidihedral and generalized quaternion groups (all of which are 2-generator groups).

Corollary 2.3. Let P be a finite p-group.

(1) If Aut(P) acts transitively on the maximal subgroups of P, then every proper characteristic subgroup C of P is contained in Φ(P). In particular, there is a bijection between the maximal subgroups of P and P/C and the kernel of the natural map Aut(P) → Aut(P/C) is a p-group.

(2) Let p be odd. If P ≅ C_{p^{m}} × C_{p^{n}} and there exists a subgroup A of Aut(P)

such that |A| is divisible by p and O_{p}(A) = 1, then m = n. Assume that p = 2.

(3) If P possesses a self-centralizing subgroup isomorphic to V_{4}, then P is di-

hedral or semidihedral. Assume further that Aut(P) is not a 2-group.

(4) If P ≅ C_{2^{m}} × C_{2^{n}}, then m = n.

(5) If P has maximal class, then P ≅ Q_{8}.

(6) If P has a cyclic maximal subgroup, then P ≅ V_{4} or P ≅ Q_{8}.

Proof. (1) Let C be a proper characteristic subgroup of P. As it is contained in some maximal subgroup and is fixed by all automorphisms of P, it is contained in all maximal subgroups and hence is contained in Φ(P). This implies the bijection between maximal subgroups and the final statement is an immediate consequence of Burnside’s result, [10, Theorem 5.1.14].

(2) As C_{p^{m}} × C_{p^{n}} has maximal subgroups C_{p^{m}} × C_{p^{n−1}} and C_{p^{m−1}} × C_{p^{n}}, the result follows from Proposition 2.1.

(3) By [14, Satz III.14.23], P has maximal class. Furthermore, P cannot be a generalized quaternion group as it contains more than one involution.

(4) The proof is the same as in (2) but utilizing Corollary 2.2.

(5) Any maximal-class 2-group except Q_{8} has a pair of nonisomorphic maximal subgroups and so the result follows from Corollary 2.2.

(6) Any such P is a 2-generator group. If P is abelian, then by Corollary 2.2 and part (4), P ≅ V_{4}. If P is nonabelian, then by [10, Theorem 5.4.4], P has maximal class or is a modular 2-group. The latter case is impossible since every modular 2-group has at least two nonisomorphic maximal subgroups (see Section 3.1 for the definition of modular 2-groups and this statement). The result now follows from (5).

□

In the case where the homocyclic abelian subgroup has exponent greater than 2, we can also pin down the structure of the ambient 2-group.

Lemma 2.4. Let F be a saturated fusion system on a finite 2-group P and let Q be a proper, abelian, F-centric, F-radical subgroup of P.

(1) If |Q| ≤ 4, then Q ≅ V_{4} and P is dihedral or semidihedral.

(2) If |Q| > 4, and P has 2-rank 2, then |P : Q| = 2 and P is wreathed.

Proof. Since Q is F-centric and F-radical, Aut(Q) is not a 2-group. In particular, Q is not cyclic. Therefore, by Corollary 2.3.3, if Q ≅ V_{4}, then P is dihedral or semidihedral. So, we may assume that |Q| > 4. By Corollary 2.3.4, Q ≅ C_{2^{n}} ×C_{2^{n}}. Let R = N_{P} (Q) > Q. Since Q is F-radical, Aut_{F}(Q) ≅ S_{3} and |R : Q| = 2. Let x and y be generators of Q, and let t be an element of R \ Q; considering the action of Aut_{F}(Q) on Z \ {1} (on which it must act faithfully), we see that x^{t} ∉ ⟨x⟩, and so we may choose y = x^{t}. To prove that R is a wreathed 2-group, we need to show that R \ ⟨x, y⟩ contains an element of order 2.

Since t^{2} ∈ Q, and t centralizes t^{2}, write t^{2} = (xy)^{i}. We have

(x^{−i}t)^{2} = x^{−i}t^{2}(t^{−1}x^{−i}t) = x^{−i}x^{i}y^{i}y^{−i} = 1,

so that R is wreathed. Since Q is the unique maximal abelian subgroup of R, it is characteristic in R and hence Q ⊴ N_{P} (R). Thus P = R, as required. □

3. Metacyclic 2-Groups

3.1. Modular 2-groups. Wong [29] proved that any finite group with a modular 2-group as a Sylow 2-subgroup is 2-nilpotent. Recently, Ensslen and Külshammer [9, Theorem 7] proved a version of Wong’s results for blocks, namely, that a block with defect group a modular 2-group is a nilpotent block.

Here we show that there is a single saturated fusion system on a modular 2-group P, namely F_{P} (P); this generalizes the above two results. Recall that the modular 2-groups have the form

Mod_{n} = ⟨x, y | x^{2^{n−1}} = 1 = y^{2}, yxy = x^{2^{n−2}+1}⟩,

where n ≥ 4. In particular, modular 2-groups are 2-generator 2-groups with 2-rank 2. The subgroup ⟨x⟩ is a cyclic subgroup of index 2, and the subgroup ⟨x^{2}, y⟩ is a noncyclic abelian subgroup of index 2, proving the assertion in the proof of Corollary 2.3.6. As a warm-up to the general theorem on metacyclic 2-groups, we prove the following, now straightforward, proposition.

Proposition 3.1. The only saturated fusion system on a modular 2-group is the trivial fusion system.

Proof. Let P = Mod_{n} and let F be a saturated fusion system on P. As P has nonisomorphic maximal subgroups, Corollary 2.2 implies that Aut(P) is a 2-group. Moreover, as every proper subgroup of P is abelian and P is not dihedral, semidihedral or wreathed, Lemma 2.4 implies that P has no F-centric, F-radical subgroups. By Alperin’s fusion theorem and the Sylow axiom for fusion systems, F is trivial. □

3.2. Automorphisms of metacyclic subgroups. We need to understand how p′-automorphisms act on metacyclic p-groups. For p = 2 we obtain a complete description of which metacyclic p-groups afford p′-automorphisms, and for odd p we determine which metacyclic p-groups have an automorphism group consistent with being a centric radical subgroup in a larger p-group.

Proposition 3.2. Let Q be a metacyclic p-group such that Aut(Q) is not a p-group. (1) If p = 2, then Q ≅ C_{2^{n}} × C_{2^{n}} or Q ≅ Q_{8}.

(2) If p is odd and A is a subgroup of Out(Q) such that |A| is divisible by p and O_{p}(A) = 1, then Q ≅ C_{p^{n}} × C_{p^{n}}.

Proof. If Q is abelian then the result holds by Corollary 2.3.4, so we assume that Q is nonabelian. As Q′ is cyclic, the unique involution of Q′ generates a characteristic subgroup Z of Q of order 2. Since Z ≤ Φ(Q), we see that Aut(Q/Z) is not a 2-group, so by induction Q/Z ≅ Q_{8} or Q/Z is homocyclic abelian.

In the former case, |Q/Q′| = |(Q/Z)/(Q′/Z)| = |Q_{8}/Q′_{8}| = 4 and so, by [10, Theorem 5.4.5], Q is dihedral, semidihedral or generalized quaternion. As none of these has Q_{8} as a proper quotient by the center, we derive a contradiction. If Q/Z is homocyclic abelian, then the preimage in Q of the subgroup generated by all involutions in Q/Z is a characteristic subgroup C of Q of order 8 containing all involutions of Q. If Aut(C) is a 2-group, then an automorphism of odd order in Aut(Q) induces the trivial automorphism on C, so in particular it induces the trivial automorphism on Ω_{1}(Q/Z), contradicting [10, Theorem 5.2.4], which states that a p′-automorphism of an abelian p-group cannot act trivially on the elements of order p. Hence Aut(C) is not a 2-group, and so C ≅ Q_{8}. This implies that Q has a unique involution; in particular, any abelian subgroup of Q is cyclic and so, by [10, Theorem 5.4.10(ii)], Q is a generalized quaternion group. Corollary 2.3.5 implies that Q ≅ Q_{8}, proving (1).

Suppose that Q is nonabelian. Since Q′ is cyclic, Z = Ω_{1}(Q′) has order p; the hypotheses are inherited by Q/Z, so we may assume by induction that Q/Z is homocyclic abelian, of order p^{2n}. It is well known (see for example [23, Lemma 1.2]) that Q possesses a unique elementary abelian subgroup R of order p^{2}, and since this is contained in one maximal subgroup, it is contained within all of them, as they are all isomorphic by Proposition 2.1. Again, the hypotheses are inherited by Q/R since R ≤ Φ(P), and so Q/R is homocyclic abelian, of order p^{2m}. However, now |Q| = p^{2n+1} = p^{2m+2}, a contradiction. Hence, Q is abelian, and 2 follows from Corollary 2.3.2. □

In [20], Linckelmann and Mazza define the Dade group, D(F, P), of a saturated fusion system F on a finite p-group P as the F-stable elements in the Dade group, D(P), of P. Furthermore, they give a criterion for when D(F, P) = D(P).

Theorem 3.3 ([20, Theorem 6.1]). Let F be saturated fusion system on a finite 2-group P. If every F-essential subgroup of P is isomorphic to V_{4} or Q_{8}, then D(F, P) = D(P).

This is used to prove that D(F, P) = D(P) whenever p = 2 and P is of maximal class. We will use our preparatory work to generalize their result to all metacyclic 2-groups.

Corollary 3.4. Let F be a saturated fusion system on a finite 2-group P. If P is metacyclic, then D(F, P) = D(P).

Proof. As P is metacyclic, it is not wreathed and so, by Lemma 2.4, any abelian F-essential subgroup of P is isomorphic to V_{4}. Furthermore, by Proposition 3.2.1, every nonabelian F-essential subgroup is isomorphic to Q_{8}. The result now follows from Theorem 3.3. □

3.3. Fusion systems on metacyclic p-groups. If F is a saturated fusion system on a finite p-group P and Z is a central subgroup of F, i.e., F = C_{F}(Z), then the following result of Kessar and Linckelmann gives a handy way of passing between the centric radical subgroups of F and F/Z. We will use it in the proof of Theorem 3.7.

Proposition 3.5 ([16, Proposition 3.1]). Let F be a saturated fusion system on a finite p-group P and let Z be a subgroup of P such that F = C_{F}(Z). If Q is an F-centric, F-radical subgroup of P, then Q/Z is an F/Z-centric, F/Z-radical subgroup of P/Z.

Lemma 3.6. Let F be a saturated fusion system on a finite p-group P. If the proper F-centric, F-radical subgroups of P have a common center Z, then Z is strongly F-closed.

Proof. Let T ≤ Z, let Q be a proper F-centric, F-radical subgroup of P containing T and let φ ∈ Aut_{F}(Q). As T ≤ Z = Z(Q) and Z(Q) is characteristic in Q, we have φ(T) ≤ φ(Z(Q)) = Z(Q) = Z. □

The following theorem strengthens Proposition 3.1, showing that the only metacyclic 2-groups that give rise to nontrivial saturated fusion systems are homocyclic, dihedral, semidihedral or generalized quaternion. At the end of Section 5, we will construct all saturated fusion systems on these groups. Benjamin Sambale independently proved this result using different methods in [25, Theorem 1].

Theorem 3.7. Let F be a saturated fusion system on a finite metacyclic 2-group P. If P is not homocyclic abelian, dihedral, semidihedral or generalized quaternion, then F is trivial.

Proof. Let Q be an F-essential subgroup of P. If Q is abelian, then Lemma 2.4 implies that P is dihedral, semidihedral, or wreathed; since wreathed 2-groups are not metacyclic, this case cannot occur.

Therefore, every proper F-essential subgroup of P is nonabelian and hence, by Proposition 3.2.1, isomorphic to Q_{8}. As each F-essential subgroup Q is F-centric, 1 ≠ Z(P) ≤ Z(Q). As the last group has order 2, Z(Q) = Z(P). By Lemma 3.6, Z = Z(P) is strongly F-closed. The unique nontrivial element of Z(P) is thus central in F. We conclude that Z = Z(F). By Proposition 3.5, Q/Z is F/Z-centric and so P/Z has a self-centralizing subgroup isomorphic to V_{4}. In particular, P has maximal class, a contradiction.

The only remaining case is where P has no proper F-essential subgroups. In this case, we invoke Proposition 3.2 to conclude that Aut(P) is a 2-group. This proves that F is trivial. □

Blocks of finite groups with dihedral, semidihedral or generalized quaternion defect groups are tame, and so this theorem yields the following immediate corollary.

Corollary 3.8. Any wild 2-block with a metacyclic, nonhomocyclic defect group is nilpotent.

Another application of our theorem to 2-blocks with metacyclic defect groups is Linckelmann’s “gluing problem” (see [19, Conjecture 4.2]) for blocks. Recently, Sejong Park showed [24, Theorem 1.2] that the gluing problem has a unique solution for tame blocks. Theorem 3.7 enables us to extend his result to all blocks with metacyclic defect groups. To state this result, we need to set up some notation.

For a saturated fusion system F on a finite p-group P, let C denote the poset of F-conjugacy classes of chains of F-centric subgroups of P (the ordering is induced by taking subchains). For i ∈ ℕ, let A_{F}^{i} denote the covariant functor sending a chain σ ∈ C to the abelian group H^{i}(Aut_{F}(σ), k^{×}). This allows us to define the cohomology of C with coefficients in A_{F}^{i}.

Külshammer and Puig showed in [17] that if F is the fusion system of a block of a finite group, then the block determines an element of H^{0}(C, A_{F}^{2}). Linckelmann conjectured that this element was in the image of the map

H^{2}(F^{c}, k^{×}) → H^{0}(C, A_{F}^{2})

and showed (see [18]) that every saturated fusion system F gives rise to the exact sequence,

0 → H^{1}(C, A_{F}^{1}) → H^{2}(F^{c}, k^{×}) → H^{0}(C, A_{F}^{2}) → H^{2}(C, A_{F}^{1}) → H^{3}(F^{c}, k^{×}),

which we will use without reference in the proof of the next corollary.

Corollary 3.9. Let F be a saturated fusion system on a finite 2-group P and let k be an algebraically closed field of characteristic 2. If P is metacyclic, then H^{2}(F^{c}, k^{×}) = H^{0}(C, A_{F}^{2}) = 0. In particular, the gluing problem for 2-blocks with metacyclic defect group has a unique solution.

Proof. If σ is a chain of F-centric subgroups in C and if H^{2}(Aut_{F}(σ), k^{×}) = 0, then A_{F}^{2} = 0 and hence H^{0}(C, A_{F}^{2}) = 0. This implies that H^{1}(C, A_{F}^{1}) ≅ H^{2}(F^{c}, k^{×}) and it would then remain to show that H^{1}(C, A_{F}^{1}) = 0. By [24, Theorem 1.2] and Theorem 3.7, it will suffice to consider the cases where either F = F_{P} (P) or P ≅ C_{2^{n}} × C_{2^{n}}. In the former case A_{F}^{1} = A_{F}^{2} = 0 which implies that H^{1}(C, A_{F}^{1}) ≅ H^{2}(F^{c}, k^{×}) = 0 giving the result.

If P ≅ C_{2^{n}}×C_{2^{n}}, then there is only one F-centric subgroup, namely P, and hence H^{1}(C, A_{F}^{1} ) = 0. Furthermore, by Theorem 5.3, if F ≠ F_{P} (P), then F = F_{P} (P ⋊C_{3}) and hence Aut_{F}(P) ≅ C_{3}. As, H^{2}(C_{3}, k^{×}) = 0, the proof is complete. □

We now deal with the odd primes case, giving a structural proof of Stancu’s result from [26].

Theorem 3.10. Let F be a saturated fusion system on a finite p-group P with p odd. If P is metacyclic, then F = N_{F}(P).

Proof. Let P be a minimal counterexample and let F be a saturated fusion system on P such that N_{F}(P) ≠ F. If P is abelian, the extension axiom implies every automorphism of a fully F-normalized subgroup extends to P and hence F = N_{F}(P), so P is nonabelian. By Alperin’s fusion theorem and Proposition 3.2.2, P contains an F-essential subgroup Q ≅ C_{p^{n}} × C_{p^{n}} for some n ∈ ℕ. In fact, by [23, Lemma 2.1], Q is the unique homocyclic abelian subgroup of P of order p^{2n} and hence Q is characteristic in P. Since Q is fully F-normalized, Aut_{P} (Q) is a Sylow p-subgroup of Aut_{F}(Q). As this latter group is isomorphic to a subgroup of GL_{2}(p),

|P : Q| = |N_{P} (Q) : C_{P} (Q)| = |Aut_{P} (Q)| = p,

and hence |P | = p^{2n+1}.

If n = 1, then P is an extraspecial group of order p^{3}. As Q ≅ C_{p} × C_{p} contains exactly p^{2} − 1 elements of order p, P must have exponent p^{2}. As Q is F-essential, the proof of Proposition 2.1 implies that Aut_{F}(Q) = Out_{F}(Q) contains SL_{2}(p) and hence contains the automorphism ϕ that inverts all elements of Q. Since ϕ centralizes Aut_{P} (Q), the extension axiom gives an extension ψ ∈ Aut_{F}(P) of ϕ, which may be chosen to have order 2. Since P has p cyclic subgroups of index p, ψ must fix (at least) one of them; call it R. The center of P is contained in R and ψ inverts Z(P). Thus, ψ induces the unique automorphism of order 2 on R (which is, of course, inversion). Therefore, ψ inverts all elements of QR = P, a contradiction since P in nonabelian.

We conclude that n > 1. As P/Φ(Q) is metacyclic, the minimality of P implies that P/Φ(Q) is resistant. Furthermore, since Q is the unique F-essential subgroup of P and since Φ(Q) is characteristic in P, it follows that Φ(Q) is strongly F-closed. Thus, \overline{F} = F/Φ(Q) is a saturated fusion system on \overline{P} = P/Φ(Q).

Since \overline{P} is resistant, it is the only \overline{F}-centric, \overline{F}-radical subgroup. In particular, \overline{Q} is not \overline{F}-centric and \overline{F}-radical. However, since \overline{Q} is elementary abelian, the kernel of the canonical surjection Aut_{F}(Q) → Aut\overline{F}(\overline{Q}) is a normal p-subgroup. It is, therefore, trivial since Q is F-radical. Thus, the above surjection is an isomorphism and \overline{Q} is \overline{F} -radical. It follows that \overline{Q} cannot be \overline{F} -centric. But, as [\overline{P} : \overline{Q}] = p, this implies C\overline{P} (\overline{Q}) = \overline{P}, i.e., \overline{Q} is contained in the center of \overline{P}. Therefore, \overline{P} is abelian. This is impossible, however, as then Aut\overline{P} (\overline{Q}) is trivial while p | |Aut\overline{F}(\overline{Q})|, contradicting the saturation of \overline{F}.

□

4. Suzuki 2-groups

In this short section we discuss Suzuki 2-groups and show that they are always resistant. This result is needed in the following section.

Definition 4.1. A finite 2-group is a Suzuki 2-group if P is nonabelian, possesses more than one involution, and has an odd-order automorphism ϕ such that ϕ transitively permutes the elements of Ω_{1}(P).

Examples of Suzuki 2-groups are the Sylow 2-subgroups of the Suzuki simple groups ^{2}B_{2}(q), as well as the Sylow 2-subgroups of the groups PSL_{3}(2^{n}) and PSU_{3}(2^{n}). The main theorem on these groups is a result of Graham Higman.

Theorem 4.2 (Higman [13]). If P is a Suzuki 2-group, then

Ω_{1}(P) = Z(P) = Φ(P) = P′,

and so P has exponent 4 and class 2.

In order to prove our result, we use the following result of Aschbacher.

Proposition 4.3 (Aschbacher). Let F be a saturated fusion system on a finite p-group P. If Q is a strongly F-closed subgroup of P, then F = N_{F}(Q) if and only if Q possesses a central series of strongly F-closed subgroups

1 = Q_{0} ≤ Q_{1} ≤ ⋯ ≤ Q_{n} = Q.

Proof. See [7, Proposition 3.4]. □

Theorem 4.4. Suzuki 2-groups are resistant.

Proof. Let P be a Suzuki 2-group. Since Q = Ω_{1}(P) is elementary abelian and contains all involutions, it is strongly F-closed. Therefore, the sequence 1 ≤ Q ≤ P is a central series of strongly F-closed subgroups. Hence, by Proposition 4.3, F = N_{F}(P), giving us the result. □

5. Center-free and simple saturated fusion systems of 2-rank 2

Using the results of the previous sections, we classify center-free saturated fusion systems on 2-groups of 2-rank 2. In [4], it was shown that the quotient of a saturated fusion system F by its center preserves structure, in the sense that one may reconstruct F, up to isomorphism, from F/Z(F) and the original p-group on which F is defined. Iterating this procedure results in a saturated fusion system with trivial center. Thus, understanding center-free saturated fusion systems is of considerable importance. For instance, Corollary 6.14 in [4] says that F is an exotic fusion system if and only if F/Z(F) is exotic. At the end of the section, we classify all simple saturated fusion systems on 2-groups of 2-rank 2.

Let P be a finite 2-group of 2-rank 2 and let F be a center-free saturated fusion system on P. Since Z(F) = 1, there is some F-centric, F-radical subgroup U and an automorphism ϕ such that Ω_{1}(Z(U)) ≰ C_{U}(ϕ). In particular, Ω_{1}(Z(U)) has order 4 and therefore contains all involutions in U.

If U is abelian, then Lemma 2.4 implies that P is dihedral, semidihedral, wreathed or homocyclic abelian. If U is nonabelian then U is a Suzuki 2-group, and by [11, Lemma 2.2.6], U is the Sylow 2-subgroup of PSU_{3}(4), a Suzuki 2-group of order 2^{6}. Suppose that U < P; since U is F-centric, the fusion system N_{F}(U) is constrained; if G is a finite group whose fusion system is that of N_{F}(U), then N_{P} (U) has 2-rank 3 by [11, Lemma 5.2.9(c)]. Since P has 2-rank 2, we conclude that P = U, giving the following theorem.

Theorem 5.1. Let F be a saturated fusion system on a finite 2-group P with 2rank 2. If Z(F) = 1, then P is dihedral, semidihedral, wreathed, homocyclic abelian, or the Sylow 2-subgroup of PSU_{3}(4).

It is easy to find a center-free fusion system on each of these groups: the simple groups PSL_{2}(q), PSL_{3}(q), PSU_{3}(q) (for q odd), and PSU_{3}(4) give examples for all but the homocyclic abelian, and in this case the group (C_{2^{n}} × C_{2^{n}}) ⋊ C_{3} is an example. We next construct all saturated fusion systems on the groups mentioned in this result.

Definition 5.2. Let F be a saturated fusion system on a finite p-group P. The essential rank of F, denoted by rk_{e}(F), is the number of F-conjugacy classes of F-essential subgroups of P.

Theorem 5.3. Let F be a saturated fusion system on a finite 2-group P. If P is dihedral, semidihedral, generalized quaternion or wreathed, then rk_{e}(F) ≤ 2 and F is isomorphic to the fusion system on P of one of the groups in the following table:

rk_{e}(F)  0  1  2
D_{2^{n}}  F_{P} (P)  PGL_{2}(q)  PSL_{2}(q)
q ≡ ±1 mod 8
SD_{2^{n}} PSL_{2}(q^{2}) ⋊ C_{2}, GL_{2}(q) PSL_{3}(q)
q ≡ 3 mod 4 q ≡ 3 mod 4
Q_{2^{n}} SL_{2}(q).C_{2} SL_{2}(q)
q odd
C_{2^{n}} ≀ C_{2} (C_{2^{n}} × C_{2^{n}}) ⋊ S_{3}, GL_{2}(q) PSL_{3}(q)
q ≡ 1 mod 4 q ≡ 1 mod 4

If P is homocyclic abelian or the Sylow 2-subgroup of PSU_{3}(4), then rk_{e}(F) = 0 and F is isomorphic to the fusion system on P of one of the groups in the following table:

\begin{matrix}C_{2^{n}} × C_{2^{n}} & Suz \\ F_{P} (P), F_{P} (P ⋊ C_{3}) & F_{P} (P), F_{P} (P ⋊ C_{3}), F_{P} (P ⋊ C_{5}), F_{P} (P ⋊ C_{15}) \end{matrix}

Proof. The proof of this theorem is broken up into three parts: the dihedral, semidihedral and quaternion groups, wreathed groups, and the homocylic abelian and Suzuki groups.

Dihedral, Semidihedral and Quaternion 2-groups. The saturated fusion systems on D_{2^{n}}, SD_{2^{n}} and Q_{2^{n}} are quite similar. We will try, as much as possible, to proceed in parallel for all three classes of groups.

By Lemma 2.4 and Proposition 3.2.1, any F-essential subgroup of P is isomorphic to V_{4} or Q_{8} and has F-outer automorphism group isomorphic to S_{3}. In any of the cases, |Z(P)| = 2 and Z(P) is contained in every F-essential subgroup of P. By Alperin’s fusion theorem and the fact that the automorphism groups of dihedral, semidihedral and quaternion 2-groups are again 2-groups, F is completely determined by the automorphism groups of the F-essential subgroups. If rk_{e}(F) = 0, then F = F_{P} (P) for any of our potential P, so we may assume that there is at least one F-essential subgroup.

If P is dihedral, then the F-essential subgroups must all be isomorphic to V_{4}. The dihedral groups have two conjugacy classes of V_{4}-subgroups (that are Out(P)- conjugate) and three conjugacy classes of involutions. An automorphism of order 3 of V_{4} permutes its three involutions, and as one of these involutions is central in P, two of the conjugacy classes of involutions in P will fuse in F. If one or both classes of V_{4} subgroups are essential then there are two or a single F-conjugacy class of involutions respectively. In the former case this is the fusion system of PGL_{2}(q) (for the appropriate odd q) and in the latter case this is the fusion system of PSL_{2}(q) (again, for the appropriate odd q).

If P is generalized quaternion, then every F-essential subgroup is isomorphic to Q_{8}. As P has a unique involution, Z(F) = Z(P) ≠ 1. Thus, F/Z(F) is a fusion system on a dihedral group and so by [4, Corollary 6.14], F is the fusion system of a finite group. In fact, it is a central extension of a fusion system of a dihedral group.

Finally, if P is semidihedral, then there are three maximal subgroups: a dihedral subgroup, D, a cyclic subgroup, C, and a generalized quaternion subgroup, Q. The two D-conjugacy classes of subgroups isomorphic with V_{4} are fused in P, as are the two Q-conjugacy classes of subgroups isomorphic to Q_{8}. (If |P | = 16, there is a unique subgroup of P isomorphic to Q_{8}.) Therefore there is a single P -conjugacy class of subgroups isomorphic with V_{4}, and similarly for subgroups isomorphic with Q_{8}.

[figure: P; D; C; Q; V_{1}; V_{2}; Q_{1}; Q_{2}]

We conclude that there are at most two F-conjugacy classes of F-essential subgroups of P, one isomorphic to V_{4} and one isomorphic to Q_{8}. As there are two conjugacy classes of involutions in a semidihedral group, there is an F-essential isomorphic to V_{4} if and only if the involutions form a single F-conjugacy class. Semidihedral groups also have precisely two conjugacy classes of elements of order 4 (and these are permuted by an automorphism of order 3 of Q_{8}). Thus, there is an F-essential subgroup isomorphic to Q_{8} if and only if the elements of order 4 form a single F-conjugacy class. If both of these situations occur, our fusion system comes from PSL_{3}(q). If only the latter situation occurs, we get GL_{2}(q). Finally, if only the former situation occurs, then we get the fusion system coming from a nonsplit extension of PSL_{2}(q^{2}) by the field automorphism. (This group has a unique conjugacy class of involutions but must have more than one class of elements of order 4 since some lie outside of PSL_{2}(q^{2}).)

Wreathed 2-groups. Let P be the wreathed group C_{2^{n}} ≀ C_{2} for some n ≥ 2 (as n = 1 gives P ≅ D_{8}), and let Q denote the base group C_{2^{n}} × C_{2^{n}} generated by a and b, a characteristic subgroup of P with |P : Q| = 2. Since P is 2-generator, and one, but not all maximal subgroups of P are abelian (e.g., C_{2^{n−1}} ≀ C_{2} is a subgroup of P), Aut(P) is a 2-group by Corollary 2.2. Hence it suffices, as in the previous case, to determine all possible essential subgroups of P. Note that P has a cyclic center of order 2^{n}, generated by ab and lying inside Q.

Fix an element t in P \ Q with a^{t} = b and let S be a subgroup of P with C_{P} (S) ≤ S such that Aut(S) not a 2-group. In [1, Lemma 1.3], it is stated that S is either Q or P -conjugate to the central product R of Z(P) and ⟨(ab^{−1})^{2^{n−2}}, t⟩ ≅ Q_{8}. Since no proof is offered there we sketch a proof here. If S is abelian then S is homocyclic abelian and contains Z(P), so that S = Q. Hence, S is nonabelian, and contains Z(P). The quotient P/Z(P) is dihedral of order 2^{n+1}, and since any automorphism of odd order acts trivially on Z(P), Aut(S/Z(P)) cannot be a 2group. Hence S/Z(P) ≅ V_{4}, and there are two conjugacy classes of such subgroups; write R and T for preimages in P of representatives of these subgroups. They are generated by the central element of P/Z(P), namely Z(P)a^{2^{n−1}}, and either Z(P)t or Z(P)at, and so the two subgroups of interest are

R = ⟨ab, a^{2^{n−1}}, t⟩ and T = ⟨ab, a^{2^{n−1}}, at⟩ = ⟨a^{2^{n−1}}, at⟩.

As the cyclic subgroup ⟨at⟩ and the noncyclic subgroup ⟨ab, a^{2^{n−1}}⟩ are both maximal in T, there is no automorphism of T of order 3. This leaves R, which is a central product of Z(P) and a Q_{8} subgroup, as claimed.

There are therefore four saturated fusion systems on P, depending on whether one of the subgroups Q, R, both Q and R, or neither Q nor R, are essential subgroups. These are the fusion systems of (respectively): (C_{2^{n}} × C_{2^{n}}) ⋊ S_{3} (with S_{3} acting by permutations on a, b and a^{−1}b^{−1} where a and b are as above); the centralizer of an involution in PSL_{3}(q) for q ≡ 1 mod 4, i.e., GL_{2}(q); PSL_{3}(q) for q ≡ 1 mod 4 (and PSU_{3}(q) for q ≡ 3 mod 4); and P itself.

Resistant Cases. The other groups mentioned above are the homocyclic abelian groups P_{n} = C_{2^{n}} ×C_{2^{n}}, and the Sylow 2-subgroup of PSU_{3}(4), which we will denote by Suz. These 2-groups are resistant (in the first case since all abelian p-groups are resistant and in the second case by Theorem 4.4), and so to determine all saturated fusion systems on these groups it suffices to determine Out(P_{n}) and Out(Suz).

In the first case, Out(P_{n}) has order 2^{α} · 3 for some α by Proposition 2.1, and so there are two saturated fusion systems on P_{n}, namely those of P_{n} and P_{n} ⋊ C_{3}.

In the second case, it is easy to see (see the following section) that Out(Suz) has automorphisms of order 3 and 5, and (up to raising by powers) there are unique such automorphisms. Hence there are four saturated fusion systems on Suz, namely those of the groups Suz, Suz ⋊ C_{3}, Suz ⋊ C_{5} and Suz ⋊ C_{15}. □

We finish the section by finding all simple fusion systems on 2-groups of 2-rank 2. Recall that if F is a saturated fusion system on a finite p-group P and if E is a saturated subsystem of F on a strongly F-closed subgroup Q of P, then E is weakly normal in F if for all subgroups S ≤ T ≤ Q and for all ϕ ∈ Hom_{F}(T, Q), conjugation by ϕ induces a bijection Hom_{E}(S, T) → Hom_{E}(Sϕ, T ϕ). A saturated fusion system is simple if its only weakly normal subsystems are the fusion system of the trivial group and itself. Note that while this definition of simplicity differs, a priori, from that given by Aschbacher in [3], recent work of the first author [6] shows that these definitions are, in fact, equivalent.

Theorem 5.4. Let F be a saturated fusion system on a finite 2-group P with 2rank 2. If F is simple, then P is dihedral, semidihedral or wreathed, and F is either the fusion system of PSL_{2}(q) for q ≡ ±1 mod 8 or PSL_{3}(q) for q odd.

Proof. By [26, Proposition 6.2], O_{p}(F) = 1. Therefore, Z(F) = 1 and we may apply Theorem 5.1, implying that P is dihedral, semidihedral, wreathed, homocyclic abelian or the Sylow 2-subgroup of PSU_{3}(4). However, P is not homocyclic abelian since the only simple fusion system on an abelian p-group is F_{C_{p}}(C_{p}) and P is not the Sylow 2-subgroup of PSU_{3}(4) since in this case, by Theorem 4.4, P is normal in F. The result now follows from Theorem 5.3 and the fact that a simple fusion system coming from a finite group must come from a simple group. □

6. Odd-order automorphisms of 2-groups

In this section we will determine all 2-groups of 2-rank 2 possessing a nontrivial odd-order automorphism, the first step in classifying all saturated fusion systems on 2-groups of 2-rank 2, which would complete the project in [8].

In order to state the main theorem of this section, we introduce the following notation for several classes of 2-groups. Let Q_{n,m} = Q_{2^{n}}×Q_{2^{m}}, QC_{n,m} = Q_{2^{n}}×C_{2^{m}} and QD_{n,m} = Q_{2^{n}} × D_{2^{n}}. The notation QC_{n,m}^{∗} and QD_{n,m}^{∗} are defined similarly but as central products instead of direct products. The first two of these series have exactly three involutions and, if n = 3, each of these possesses an automorphism of order 3 acting in the obvious way on the Q_{8}-factor and trivially on the other factor; the group Q_{3,3} possesses another automorphism of order 3 acting nontrivially on both factors. We also write C_{n,m} = C_{2^{n}} × C_{2^{m}}. The Sylow 2-subgroup of PSU_{3}(4) will again make an appearance as the only Suzuki 2-group with three involutions, and here we will again refer to it as Suz.

There are two more infinite series of groups to define; both are (nonsplit) extensions of Q_{8} × C_{2^{n}} by C_{2}. Let Q_{8} be generated by a and b, and let the cyclic factor be generated by c. The groups X_{n} and Y_{n} (of order 2^{n}) have a normal subgroup QC_{3,n−4} generated by a, b and c, with d outside having order 4 and commuting with a and b, and moving c. In Y_{n}, the element d inverts c, and in X_{n} it sends c to c^{−1}a^{2}. In X_{n}, d^{2} is the element z of order 2 in ⟨c⟩, and in Y_{n} we have d^{2} = a^{2}z.

More formally, we have

X_{n} =⟨ a, b, c, d | a^{4} = c^{2^{n−4}} = 1, b^{2} = a^{2}, a^{−1}ba = b^{−1},
d^{2} = c^{2^{n−5}}, d^{−1}cd = c^{−1}a^{2}, [a, c] = [b, c] = [a, d] = [b, d] = 1 ⟩, and
Y_{n} =⟨ a, b, c, d | a^{4} = c^{2^{n−4}} = 1, b^{2} = a^{2}, a^{−1}ba = b^{−1},
d^{2} = a^{2}c^{2^{n−5}}, d^{−1}cd = c^{−1}, [a, c] = [b, c] = [a, d] = [b, d] = 1 ⟩.

Notice that X_{6} = Y_{6}. The automorphism of order 3 on these two groups acts in the obvious way on ⟨a, b⟩ ≅ Q_{8} and trivially on c and d.

Theorem 6.1. Let P be a finite 2-group. If P has exactly three involutions, then P possesses a nontrivial odd-order automorphism ϕ if and only if P is one of the following:

(1) QC_{3,n} for some n ≥ 1 with o(ϕ) = 3;

(2) Q_{3,n} for some n ≥ 3 with o(ϕ) = 3 (there are two different automorphisms if n = 3);

(3) C_{n,n} for some n ≥ 1 with o(ϕ) = 3;

(4) X_{n} for some n ≥ 7 with o(ϕ) = 3;

(5) Y_{n} for some n ≥ 6 with o(ϕ) = 3;

(6) the Suzuki 2-group Suz with o(ϕ) = 3 or o(ϕ) = 5.

Theorem 6.2. Let P be a finite 2-group of 2-rank 2. If P has more than three involutions, then P possesses an odd-order automorphism if and only if P is one of the following:

(1) Q_{8} ∗ C_{2^{n}} for some n ≥ 2 with o(ϕ) = 3;

(2) Q_{8} ∗ D_{2^{n}} for some n ≥ 3 with o(ϕ) = 3, or n = 3 and o(ϕ) = 5;

(3) Q_{8} ≀ C_{2} with o(ϕ) = 3.

The proofs of Theorems 6.1 and 6.2 require a few results from the literature on 2-groups and their automorphisms. We firstly need MacWilliams’s four-generator theorem.

Theorem 6.3 (MacWilliams [21]). Let P be a finite 2-group, and suppose that P has no normal, elementary abelian subgroup of order 8. Then P has sectional 2-rank at most 4.

Consequently, if P has 2-rank 2 then P is at most 4-generator. Our next preliminary result is a lemma of Mazurov, although it appears in various guises elsewhere, such as [12, Theorem 1.3].

Lemma 6.4 ([22, Lemma 2]). Let P be a finite 2-group. If P possesses an odd-order automorphism that fixes all involutions of P and acts fixed-point freely on P/Φ(P), then P is a special 2-group and Ω_{1}(P) = Φ(P), i.e., P′ = Z(P) = Φ(P) = Ω_{1}(P).

Note that when we say a group has n involutions, we mean that it has exactly n involutions.

Theorem 6.5 (Janko [15, Theorem 2.2]). Let P be a finite 2-group with three involutions. If P contains a normal subgroup W isomorphic with C_{4} × C_{4}, then C_{P} (W) is metacyclic and Ω_{2}(C_{P} (W)) = W.

The next result is well known, and we include a short proof because we need slightly more information than usual, namely that the three involutions in the maximal subgroup are all central.

Lemma 6.6. If P is a 2-group of 2-rank 2, then there is a subgroup Q of index at most 2 such that Q has three involutions, all central in Q, or P is dihedral or semidihedral, in which case there is a cyclic maximal subgroup Q.

Proof. By [10, Theorem 5.4.10], if P is not dihedral or semidihedral then P contains a noncyclic normal abelian subgroup, Y. As Y is noncyclic, it must have 2-rank 2 and since it is normal and abelian, X = Ω_{1}(Y) ≅ V_{4} is normal in P. Let Q = C_{P} (X); as Q commutes with X, the three involutions in X are the only involutions in Q. Finally, [P : Q] = [P : C_{P} (X)] = |Aut_{P} (X)| ≤ 2. □

The following trivial lemma will be useful at several points.

Lemma 6.7. Let Q be a normal subgroup of a 2-group P such that P/Q ≅ V_{4} and such that each maximal subgroup of P containing Q possesses three involutions. If Q has a single involution, then P has seven involutions. Otherwise, both Q and P have three involutions.

Proof. Every element of P lies in one of its maximal subgroups, and so if Q has a single involution, each of the maximal subgroups contains two involutions not in Q and hence P has seven involutions. If Q has three involutions, then these are all the involutions in each maximal subgroup and hence in P. □

We will denote by P the set of all pairs (P, ϕ), where P is a 2-group from Theorems 6.1 and 6.2, and ϕ is a nontrivial odd-order automorphism of P. It is easy to determine the action of ϕ on a 2-group P: in all cases apart from C_{n,n}, Q_{3,3}, Q_{8} ≀ C_{2}, QD_{3,3}^{∗} with o(ϕ) = 5, and the group Suz, the group is a central product of a 2-group and the group Q_{8}. Here, the automorphism has order 3, and acts on Q_{8} in the obvious way and acts trivially on the other factor of the central product. If P is abelian then the action is obvious, and if P ≅ Q_{3,3} then we get the action just described, and also an automorphism acting diagonally on both factors; this automorphism lifts to an automorphism of Q_{8} ≀ C_{2}, which describes this case. The final cases are QD_{3,3}^{∗} and Suz, which are two small groups and hence amenable to computation via Magma.

Using a computer, we verify Theorems 6.1 and 6.2 for groups of order up to 2^{9}. This will avoid any complications due to the orders of groups being small.

Proposition 6.8. Let P be a finite 2-group of order at most 2^{9} and suppose that P has 2-rank 2. If Aut(P) is not a 2-group, then P is on the lists given in Theorems 6.1 and 6.2.

Proof. This is an easy verification using the computer algebra package Magma. The exact entries in the SmallGroup database are as in Table 6.1. □

We use the following elementary, but useful, result in the proofs of Proposition 6.10, Lemma 6.11 and Proposition 6.15.

Proposition 6.9. Let V be a 4-dimensional vector space over 𝔽_{2} and let g ∈ GL(V).

(1) If o(g) = 3 then either g acts fixed point freely, or g fixes three points that lie in a subspace of dimension 2 (i.e., they are not linearly independent).

Three involutions

Order  SmallGroup  Isomorphism Types
16  2, 12  C_{2,2}, QC_{3,1}
32  26  QC_{3,2}
64  2, 126, 238, 239, 245  C_{8,8}, QC_{3,3}, Y_{6}, Q_{3,3}, Suz
128  914, 2095, 2114, 2127  QC_{3,4}, X_{7}, Q_{3,4}, Y_{7}
256 39, 6647, 26918, 26937, 26950 C_{4,4}, QC_{3,5}, X_{8}, Q_{3,5}, Y_{8}
512 60795, 420460, 420479, 420492  QC_{3,6}, X_{9}, Q_{3,6}, Y_{9}

More than three involutions

Order  SmallGroup  Isomorphism Types
16  13  QC_{3,2}^{∗}
32  38, 50  QC_{3,3}^{∗}, QD_{3,3}^{∗}
64  185, 259  QC_{3,4}^{∗}, QD_{3,4}^{∗}
128  937, 990, 2149  Q_{8} ≀ C_{2}, QC_{3,5}^{∗}, QD_{3,5}^{∗}
256  6725, 26972  QC_{3,6}^{∗}, QD_{3,6}^{∗}
512  60897, 420514  QC_{3,7}^{∗}, QD_{3,7}^{∗}

Table 6.1. Table of small groups

(2) If o(g) = 5 then g acts fixed point freely.

Proof. Notice that GL_{4}(2), the set of all linear transformations of V, is isomorphic with A_{8}. Hence there are two conjugacy classes of elements of order 3 and a single conjugacy class of elements of order 5. Let v_{1}, v_{2}, v_{3} and v_{4} form a basis for V. If g has order 3 then there are two possibilities: either g acts by sending v_{1} ↦ v_{2}, v_{2} ↦ v_{3}, v_{3} ↦ v_{1} and fixing v_{4}, or by sending v_{i} ↦ v_{i+1} and v_{i+1} ↦ v_{i} + v_{i+1}, for i ∈ {1, 3}. The latter action has no fixed points while the former has three: v_{4}, v_{1} + v_{2} + v_{3} and v_{1} + v_{2} + v_{3} + v_{4}; these form the three nonzero elements of a 2-dimensional subspace. If g has order 5, then g (is conjugate to a map that) sends v_{i} to v_{i+1} for i ∈ {1, 2, 3} and sends v_{4} to v_{1} + v_{2} + v_{3} + v_{4}; this map has no fixed points. □

The remainder of the proof is structured as follows. We first examine the case where ϕ fixes no maximal subgroup of P. Note that there is a ϕ-invariant bijection between the subspaces of P/Φ(P) with dimension k and those with codimension k since ϕ is conjugate to an orthogonal transformation, which commutes with taking orthogonal complements. In particular, ϕ fixes no maximal subgroup of P if and only if ϕ acts fixed-point freely on P/Φ(P). We use this several times in the next proof in order to employ Lemma 6.4. We then attack the situation where there is a ϕ-invariant maximal subgroup of P in four steps: reduce to the case where P is 3-generator or 4-generator and o(ϕ) = 3; prove the result when P is 3-generator; prove the result when P is 4-generator and has a 3-generator ϕ-invariant maximal subgroups; and, finally, prove that when P is 4-generator, it does have a 3-generator ϕ-invariant maximal subgroup.

Proposition 6.10. Let P be a finite 2-group with 2-rank 2 such that Aut(P) is not a 2-group. If ϕ is a nontrivial odd-order automorphism of P and no maximal subgroup of P is ϕ-invariant, then P is on the lists given in Theorems 6.1 and 6.2. Specifically, P is 2-generator or 4-generator and either

(1) o(ϕ) = 3 and P is isomorphic to C_{n,n} or Suz, or

(2) o(ϕ) = 5 and P is isomorphic to Suz or QD_{3,3}^{∗}.

Proof. If P is 2-generator then ϕ has order 3 and permutes the maximal subgroups, all of which must have three involutions by Lemma 6.6. By Lemma 6.7, P has either three or seven involutions. If it has seven, then Φ(P) has a single involution, so that P is generated by two involutions; hence P is a dihedral group, a contradiction as then Aut(P) is a 2-group (also, no dihedral 2-group has seven involutions). Hence, P has three involutions. If P is abelian, then P ≅ C_{n,n} by Corollary 2.3.4. Otherwise, P is nonabelian, and ϕ either acts transitively on the involutions, so that P is a Suzuki 2-group, or ϕ acts trivially on the involutions. In either case, P has order 16 (by Theorem 4.2 and Lemma 6.4, respectively) and so, by Proposition 6.8, is isomorphic to C_{2,2} or QC_{3,1}, a contradiction since C_{2,2} is abelian and QC_{3,1} has nonisomorphic maximal subgroups.

If P is 3-generator, then there are seven maximal subgroups. Consequently, if o(ϕ) = 3, then there is a ϕ-invariant maximal subgroup. We conclude that o(ϕ) = 7 and that ϕ transitively permutes the maximal subgroups of P. By Lemma 6.6, again, all have exactly three involutions. The automorphism ϕ also permutes the seven normal subgroups with quotient V_{4} and so, by Lemma 6.7, P has three or seven involutions. If ϕ permutes the involutions transitively, then there must be seven involutions and the involutions are all central by Lemma 6.6; this implies the 2-rank of P is greater than 2, a contradiction. We conclude that ϕ acts trivially on the involutions. However, Lemma 6.4 then implies that Ω_{1}(P) = Z(P). In particular, the involutions of P are all central, contradicting the 2-rank of P if there are seven involutions. If there are only three involutions, then |P | = 32 and, by Proposition 6.8, P ≅ QC_{3,2}, a contradiction since QC_{3,2} has no automorphism of order 7 (or, alternatively, since not all of its maximal subgroups are isomorphic).

Finally, we assume that P is 4-generator. As P possesses no ϕ-invariant maximal subgroup and there are fifteen maximal subgroups, we conclude that o(ϕ) = 3 or o(ϕ) = 5. By Lemma 6.6, P possesses a maximal subgroup Q with three, central involutions. Let R = Q ∩ Qϕ; if R has three involutions then they are centralized by both Q and Qϕ, so that |Ω_{1}(Z(P))| ≥ 4. As P has 2-rank 2, this means that either P has three involutions or R has a single involution.

Suppose first that P has three involutions. On one hand, if ϕ acts nontrivially on these three involutions (so that o(ϕ) = 3), then it permutes them transitively, so that P is a Suzuki 2-group. By Theorem 4.2, |P | = 2^{6} and so P ≅ Suz. On the other hand, if ϕ acts trivially on the involutions of P, then, by Lemma 6.4, |P | = 2^{6} and so (P, ϕ) ∈ P by Proposition 6.8. One easily sees in this case that P ≅ Q_{3,3} with o(ϕ) = 3 or that P ≅ Suz with o(ϕ) = 5.

We now assume that R has a single involution. If R is ϕ-invariant, then ϕ induces a nontrivial automorphism on R (as no subgroup of P with index 8 and containing Φ(P) is ϕ-invariant). The only group with a single involution and with nontrivial automorphisms of odd order is Q_{8}, so that P has order 2^{5}, in contradiction with Proposition 6.8; hence R is not ϕ-invariant. However, if o(ϕ) = 3, then Proposition 6.9 makes it easy to see that the intersection of Q and Qϕ is ϕ-invariant. Hence, o(ϕ) = 5.

As P is 4-generator, R cannot be cyclic, so is generalized quaternion. Since P is 4-generator and R is 2-generator, Φ(P) = Φ(R) ≅ C_{2^{n}} is cyclic. If n = 1 then P has order 2^{5}, and P ≅ QD_{3,3}^{∗} by Proposition 6.8. If n ≥ 2, then let X be the (characteristic) subgroup of index 4 in Φ(P), and let G = P/X. The group G has order 2^{6}, has 4-generators, has an automorphism of order 5, and Φ(G) is cyclic. However, examining the groups of order 2^{6} using Magma, we find that there is no such 2-group, and hence P cannot exist if n ≥ 2. □

Having dealt with the case where there are no ϕ-invariant maximal subgroups, we may now assume that there is one. The following lemma reduces us to the case where P is either 3-generator or 4-generator, and o(ϕ) = 3. Our proof uses the fact (see [10, Theorem 5.3.2]) that ϕ acts trivially on Q if and only if ϕ acts trivially on P.

Lemma 6.11. Let P be a finite 2-group of 2-rank 2 with a nontrivial odd-order automorphism, ϕ. If P has a ϕ-invariant maximal subgroup, then P is 3-generator or 4-generator and o(ϕ) = 3.

Proof. If P is 2-generator, then the three maximal subgroups of P are all stabilized by ϕ. In particular, ϕ acts trivially on P/Φ(P) and hence on P, a contradiction. Therefore, P is 3-generator or 4-generator. If o(ϕ) = 5, then P is 4-generator and, by Proposition 6.9, ϕ acts fixed-point freely on the maximal subgroups of P, contradicting the assumption of a ϕ-invariant maximal subgroup. Finally, let P be a 2-group of minimal order such that there exists an automorphism of P of order 7, and let Q denote a ϕ-invariant maximal subgroup of P. If ϕ centralizes Q, then ϕ is the identity on P, a contradiction. Therefore, ϕ induces an automorphism of order 7 on Q, contradicting the minimality of P. □

Lemma 6.12. Let P be a finite 2-group of 2-rank 2 with a nontrivial odd-order automorphism, ϕ. If P has a ϕ-invariant maximal subgroup Q, then [P, ϕ] = [Q, ϕ] and P = [Q, ϕ]C_{P} ([Q, ϕ]). Moreover, if |P | ≥ 2^{8} then [P, ϕ] is isomorphic to Q_{8}.

Proof. Set R = [Q, ϕ]. As ϕ acts trivially on P/Q and Q/R, it acts trivially on P/R and hence [P, ϕ] = R. Moreover, by [10, Theorem 5.3.5], P = RC_{P} (ϕ), and so the three subgroup lemma implies that [R, C_{P} (ϕ)] = [[Q, ϕ], C_{P} (ϕ)] = 1. Thus, C_{P} (ϕ) = C_{P} (R), proving that P = RC_{P} (R), as claimed. For the last statement, note the result holds when |P | = 2^{8} by Proposition 6.8. If |P | ≥ 2^{9} and Q has a ϕ-invariant maximal subgroup, then by induction on |P |, [P, ϕ] = [Q, ϕ] ≅ Q_{8}. If Q does not have a ϕ-invariant maximal subgroup, then Q ≅ C_{n,n} by Proposition 6.10. It follows that Q/Ω_{1}(Q) ≅ C_{n−1,n−1} and so, by induction on n, ϕ acts trivially on P/Ω_{1}(Q) (since n ≥ 4); therefore, ϕ acts trivially on P as Ω_{1}(Q) ≤ Φ(Q) ≤ Φ(P) (also since n ≥ 4). □

Proposition 6.13. Let P be a finite 2-group of 2-rank 2 with a nontrivial odd-order automorphism, ϕ, and a ϕ-invariant maximal subgroup. If P is 3-generator, then P ≅ Q_{8} ≀ C_{2}, QC_{3,n} or QC_{3,n}^{∗}.

Proof. By Proposition 6.8, the proposition is true for |P | ≤ 2^{9}. If |P | ≥ 2^{10}, Q denotes a ϕ-invariant maximal subgroup and R = [Q, ϕ], then Lemma 6.12 implies that P = RC_{P} (R) and R ≅ Q_{8}. Thus, P is a quotient of X = R × C_{P} (R) by a (central) subgroup Y of order 2. As P is 3-generator, X is either 3-generator or 4-generator (so that C_{P} (R) is either 1-generator or 2-generator), and if X is 4-generator then Y ∩ Φ(X) = 1, so that Y is a direct factor of X. In this case, C_{P} (R) ≅ C_{2} × C_{2^{n}}, and hence X ≅ Q_{8} × C_{2} × C_{2^{n}}, with Y the direct factor of order 2; this yields P ≅ QC_{3,n}. If X is 3-generator, then C_{P} (R) is cyclic and X ≅ Q_{8} × C_{2^{n}}; this implies P ≅ QC_{3,n}^{∗}, completing the proof. □

Proposition 6.14. Let P be a finite 2-group of 2-rank 2 with a nontrivial odd-order automorphism, ϕ. If P is 4-generator and has a 3-generator ϕ-invariant maximal subgroup Q, then Q has a ϕ-invariant maximal subgroup and:

(1) Q ≇ Q_{8} ≀ C_{2};
(2) if Q ≅ QC_{3,n}, then P ≅ X_{n+4}, Y_{n+4} or Q_{3,n+1};
(3) if Q ≅ QC_{3,n}^{∗}, then P ≅ QD_{3,n+1}^{∗}.

Proof. Firstly, since Q is 3-generator and ϕ does not act trivially on Q, Proposition 6.10 implies that Q has a ϕ-invariant maximal subgroup. Proposition 6.8 implies that we may assume |P | ≥ 2^{10} (in particular, this shows that Q cannot be Q_{8} ≀ C_{2}, as then |P | = 2^{8}); by Lemma 6.12, we have R = [Q, ϕ] is isomorphic with Q_{8} and P = RC_{P} (R).

Suppose that Q ≅ QC_{3,n}, so that C_{Q}(R) ≅ C_{2} × C_{2^{n}}, and let x be an element of C_{P} (R) \ C_{Q}(R). Obviously x^{2} acts trivially on R and x^{2} ∈ C_{Q}(R), an abelian group, so x^{2} ∈ Z(Q). Write R = ⟨a, b⟩, with a and b of order 4, and write S = ⟨c⟩ for the cyclic direct factor of Q. It is easy to see that S and S̄ = ⟨a^{2}c⟩ are the only two normal subgroups of Q that are cyclic of order 2^{n}, and so either S^{x} = S or S^{x} = S̄. Let z be the involution in S, and notice that ⟨a^{2}, z⟩ is a central, elementary abelian subgroup of P (as x centralizes both a^{2} and z). Hence, P has three involutions. Also, S ∩ S̄ = ⟨c^{2}⟩ is characteristic in Q, so x sends c^{2} to some power of itself. Let W = ⟨a, c^{2^{n−2}}⟩, a normal subgroup of Q isomorphic with C_{4} ×C_{4}. By Theorem 6.5, C_{Q}(W) is metacyclic and Ω_{2}(C_{Q}(W)) = W. Since |Q : C_{Q}(W)| = 2, and C_{Q}(W) is 2-generator and P is 4-generator, C_{P} (W) = C_{Q}(W). Hence, x does not centralize W, but does centralize R; thus, x inverts c^{2^{n−2}}. In particular, Z(P) = ⟨a^{2}, z⟩, so that x has order 4.

If S^{x} = S, then c^{x} = c^{α}, where α ∈ {1, −1, 2^{n−1} − 1, 2^{n−1} + 1}. The two cases α = 1 and α = 2^{n−1} + 1 cannot arise as x inverts c^{2^{n−2}}. If α = 2^{n−1} − 1 and x^{2} is one of a^{2}, a^{2}z or z, then ax, acx and cx, respectively, are involutions in P and not in Q, a contradiction. If α = −1, then x^{2} = z gives P ≅ Q_{3,n+1} and x^{2} = a^{2}z gives P ≅ Y_{n+4}. If x^{2} = a^{2} then ⟨x, a, b⟩ would be a group with a single involution, as a^{2}z and z lie outside it, but this is impossible, so this situation cannot arise.

If S^{x} = S̄, then as above, c^{x} = a^{2}c^{α}, where α ∈ {1, −1, 2^{n−1} − 1, 2^{n−1} + 1}. As before, the two cases α = 1 and α = 2^{n−1} + 1 cannot arise as x inverts c^{2^{n−2}}. If x^{2} = a^{2}, a^{2}z and z, then ax, cx and acx, respectively, is an involution not lying in Q, and so P has more than three involutions. Thus c^{x} = a^{2}c^{−1}; again, x^{2} ≠ a^{2}, and if x^{2} = z then P ≅ X_{n+4}. If x^{2} = a^{2}z then (ax)^{2} = z and we again get X_{n+4}. This completes the proof of (1).

Finally, suppose that Q ≅ QC_{3,n}^{∗}. Here, C_{Q}(R) is cyclic of order 2^{n}, so that C_{P} (R) contains a cyclic maximal subgroup. If C_{P} (R) is cyclic itself then P is 3generator, and if C_{P} (R) is dihedral then P ≅ QD_{3,n+1}^{∗}, as claimed. It remains to show that C_{P} (R) cannot be C_{2^{n}} × C_{2}, semidihedral, generalized quaternion, or a modular 2-group, and to do this, in each case we will exhibit an elementary abelian subgroup of rank 3.

If C_{P} (R) contains a subgroup, S_{1}, isomorphic to Q_{8}, then let S denote the subgroup generated by the diagonal subgroup of R × S_{1} and the center of R × S_{1}, a group of order 16 whose image in R ∗ C_{P} (R) = P is elementary abelian of order 8, and this proves that if C_{P} (R) is semidihedral or generalized quaternion then P has 2-rank 3. If C_{P} (R) is C_{2^{n}} × C_{2}, then it is easy to see that P has 2-rank 3 or is 3generator (depending on whether you identify the involution in the cyclic subgroup or one outside), and if C_{P} (R) is a modular 2-group then it contains a maximal subgroup isomorphic with C_{2^{n−1}} × C_{2}, with the central involution of Mod_{n+1} being in the larger cyclic subgroup, and so the central product of this with R has 2-rank 3 as we have stated above. Thus, all parts are proved. □

Proposition 6.15. Let P be a finite 2-group of 2-rank 2 with a nontrivial odd-order automorphism, ϕ. If P is 4-generator and has a ϕ-invariant maximal subgroup, then P has a 3-generator ϕ-invariant maximal subgroup.

Proof. By Proposition 6.9, P has three ϕ-invariant maximal subgroups: call them Q_{1}, Q_{2} and Q_{3} and assume each is 4-generator.

[figure: P; 2; 16; Q_{1}; 16; Q_{2}; Q_{3}; ⋂ Q_{i}; Φ(P); Φ(Q_{1}); 2; Φ(Q_{2}); Φ(Q_{3}); X = ⋂ Φ(Q_{i})]

Firstly, the subgroups Φ(Q_{i}) are all different, else Q_{i}/Φ(Q_{1}) is elementary abelian of order 16 and P/Φ(Q_{1}) is therefore elementary abelian of order 2^{5} (since all elements of P/Φ(Q_{1}) lie in one of the Q_{i}/Φ(Q_{1}), so have order 1 or 2), contradicting the fact that P is 4-generator. Let X be the intersection of the Φ(Q_{i}), and let G = P/X. This group has the following properties.

(1) G is 4-generator, |G| = 2^{6}, and G possesses an automorphism ϕ of order 3.

(2) The three maximal ϕ-invariant subgroups of G are each 4-generator, and

have abelianization of order 16. The last statement, that |Q_{i}/Q′_{i}| = 16, follows by the fact that each Q_{i} has, by induction, a 3-generator ϕ-invariant maximal subgroup that, by Proposition 6.14, is one of the groups X_{n}, Y_{n}, Q_{3,n} and QD_{3,n}^{∗}, and examining the presentations of these groups, we see that in each case the derived subgroup and Frattini subgroup coincide.

One may use Magma to find all such groups of order 2^{6}. There are two 4generator groups G of order 2^{6} that possess an automorphism ϕ of order 3 such that the three ϕ-invariant subgroups of G are each 4-generator, given labels 192 and 194 in the SmallGroup database. The first is abelian, so in particular the Q_{i} are abelian, and while the second is nonabelian, the Q_{i} are all isomorphic and abelian. Hence there is no such group G, and thus no such group P can exist. □

This last proposition completes the 4-generator case, and so there is no counterexample to Theorem 6.1 or Theorem 6.2.

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