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On the Structure of Finite Groups with Periodic Cohomology

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On the Structure of Finite Groups with Periodic Cohomology On the structure of finite groups with periodic cohomology C. T. C. Wall December 2, 2010 Introduction It was shown in 1944 by Smith [36] that a non-cyclic group of order p2 (p prime) cannot act freely on a sphere. Hence if the group G does so act, every subgroup of G of order p2 is cyclic. Equivalent conditions on G are that every abelian subgroup of G is cyclic, and that every Sylow subgroup of G is cyclic or quaternionic: the proof of equivalence is not difficult, see e.g. Wolf [47, 5.3.2]. Another equivalent condition, due to Tate (see Cartan and Eilenberg [10, XII,11.6]) is that the cohomology of G is periodic. We shall call groups G satisfying these conditions P−groups. We will say that a finitely dominated CW complex X with fundamental group G and universal cover homotopy equivalent to SN−1 is a (G, N)-space. In this situa- tion, G is still a P−group. Homotopy types of oriented (G, N)-spaces X correspond N ˜ to cohomology generators g ∈ H (G; Z). There is an obstruction o(g) ∈ K0(ZG) to existence of a finite CW-complex homotopy equivalent to X. Such an X which is a manifold we will call a (topological or smooth) space-form. A representation ρ : G → UN of G is said to be an F−representation if the 2N−1 N induced action of G on the unit sphere S ⊂ C is free. The quotient Xρ := S2N−1/G is thus an example of a smooth space-form. In previous work on the topological spherical space-form problem [41], [29], we cited [47] for the list of P−groups. However, in [45] the need was felt for a sharper account of the classification. It is the object of this paper to provide this. Since this involves some reworking, we go back as far as possible to direct arguments. The group theory of P−groups was considered by Zassenhaus [48], who eluci- dated their structure in the soluble case. The results in the insoluble case depend on a key paper of Suzuki [37]; the complete classification was described (without full details) in Wolf [47]. The P−group condition is not quite inherited by quo- tient groups: if we enlarge the class to P0−groups by also permitting the Sylow 2-subgroup to be dihedral, it becomes so. We start with notation and preliminaries. In §2, we quote Suzuki’s result, which 0 implies that the only non-abelian simple P −groups are the P SL2(p)(p > 3 prime), and show how it follows that any P0−group G modulo its (odd) core is either a 2- group or isomorphic to some SL2(p) or TL2(p)(p ≥ 3 prime), where TL2(p) is defined below. This yields the list of types I-VI given by Wolf [47]. In §3 we give further details for P−groups, with explicit presentations in each case. The use of induction theorems (following Swan [38]) showed the need for under- standing hyperelementary subgroups of the given group. We finesse the problem of listing all such subgroups by listing in §4 subgroups which are maximal subject to having type I or II: since such groups have a non-trivial cyclic normal subgroup, this can be achieved by studying normalisers of cyclic subgroups. Those type II cases when there is not a cyclic group of index 2 in G present a more complicated representation theory, and cause certain difficulties in the study 1 of free group actions. In §4, we also analyse the general P−group G according to the existence or not of such subgroups: this refines types II, IV and VI into rtypes. A first use of our list is to calculate the period and the Artin exponent. In §5 we re-state, with a sketch of proof, Wolf’s classification of free orthogonal actions. The rtypes help clarify the case distinctions in [47, Theorem 7.2.18]. We also calculate the homotopy type of each quotient space. The finiteness obstruction is discussed in §6: most of this follows [46], but I have added references to subsequent results. The application of these results to the problem of existence of space-forms is given in §7: following the earlier papers [41], [29], [45] we are able to determine in all except certain type II cases the dimensions of the spheres on which G can act freely. Most of this is now 30 years old: we also include a survey of later results, which are mostly due to Hambleton, Madsen and Milgram. In §8 we give a corresponding discussion of classification of space-forms. I am indebted to Ian Hambleton and to Ib Madsen for helpful comments on these sections. A version of the first 4 sections of this paper, written jointly with my late friend Charles Thomas, appeared as a preprint 30 years ago. As was kindly pointed out to me, it was badly presented (my fault, not Charles’), with several minor errors; it was also interdependent with other papers. Here I have taken the opportunity to polish the original (also, the list of maximal type I and II subgroups is new, and leads to simpler proofs of several results), to include related material to make a coherent narrative, and to include the survey in the two final sections. In the course of our earlier work on this topic, we referred to Wolf’s book [47] for the necessary group theory: indeed, that book was a constant companion. It is thus a pleasure to dedicate this paper to Joe Wolf on the occasion of his 75th birthday. 1 Notation and preliminaries We will use the following notation for (isomorphism classes of) groups: × Fn denotes the ring Z/nZ for any n ∈ N, and Fn its multiplicative group of invertible elements. n Cn: cyclic of order n: hx | x = 1i. n 2 −1 −1 D2n: dihedral of order 2n: hx, y | x = y = 1, y xy = x i. n 2 4 −1 −1 Q4n: quaternionic of order 4n: hx, y | x = y , y = 1, y xy = x i. Groups Q(2k`, m, n) are defined below. GL2(p): group of invertible 2 × 2 matrices over the Galois field Fp; throughout this paper, p will be an odd prime. SL2(p): matrices in GL2(p) with determinant 1. P GL2(p): quotient of GL2(p) by its centre (the group of scalar matrices). P SL2(p): the image of SL2(p) in P GL2(p). TL2(p): let ζ be the outer automorphism of SL2(p) induced by conjugation by ω 0 w := , where ω generates ×. Then 0 1 Fp ω 0 TL (p) = SL (p), z | z−1gz = gζ for all g ∈ SL (p), z2 = . (1) 2 2 2 0 ω−1 The four latter groups are all P0−groups and form a diagram SL2(p) −→ TL2(p) ↓ ↓ , (2) P SL2(p) −→ P GL2(p) 2 where the horizontal arrows are inclusions of subgroups of index 2 and the verticals epimorphisms with kernel of order 2 (the second vertical arrow is defined by sending z to the class of w). ∗ ∗ ∗ We write T = SL2(3), O = TL2(3) and I = SL2(5) for the binary tetrahe- ∗ ∗ dral, octahedral and icosahedral groups; variants Tv and Ov are defined below. We will refer to a group C with normal subgroup A and quotient C/A ≡ B as an extension of A by B. We write Aut(G) for the group of automorphisms of a group G, Inn(G) for the (normal) subgroup of inner automorphisms, Out(G) for the quotient group. Write Z(G) for the centre of G. The maximal normal subgroup of odd order of G will be termed the core of G, and denoted O(G). We denote the order of a group G by |G|, and write Gp for a Sylow p−subgroup of G. A group is said to be p−hyperelementary if it is an extension of a cyclic group of order prime to p by a p−group. r For p prime, write νp(n) for the largest integer r with p | n. × For r, n ∈ N, write ordn(r) for the order of r in Fn . The classification of group extensions by the method of Eilenberg & MacLane [26, IV,8] proceeds as follows. Any extension C of A by B determines a homo- morphism h : B → Out(A). There is a natural restriction map r : Out(A) → Aut(Z(A)): regard Z(A) as B−module via r ◦ h. There exist extensions corre- sponding to (A, B, h) iff a certain obstruction in H3(B; Z(A)) vanishes; these are then classified by H2(B; Z(A)). If A is abelian, then Z(A) = A and the zero element of H2(B; A) corresponds to the split extension of A by B. For a split extension, the set of splittings up to conjugacy can be identified with H1(B; A). Recall that if B and A have coprime orders, all groups Hr(B; A) vanish. We apply this method to study extensions with one of the groups cyclic and the other one in (2). In each case, we need to know the centre Z(A) and the outer automorphism group Out(A). We see easily that the centre of each of SL2(p) and TL2(p) is the group {±I} of order 2; the groups P SL2(p) and P GL2(p) have trivial centre. Lemma 1.1 The groups Out(P SL2(p)) and Out(SL2(p)) have order 2, all auto- morphisms being induced by P GL2(p), resp.
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