On the structure of finite groups with periodic cohomology

Transcription

1 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 p 2 (p prime) cannot act freely on a sphere. Hence if the group G does so act, every subgroup of G of order p 2 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 S N 1 is a (G, N)-space. In this situation, G is still a P group. Homotopy types of oriented (G, N)-spaces X correspond to cohomology generators g H N (G; Z). There is an obstruction o(g) K 0 (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 U N of G is said to be an F representation if the induced action of G on the unit sphere S 2N 1 C N is free. The quotient X ρ := S 2N 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 elucidated 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 quotient groups: if we enlarge the class to P 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 implies that the only non-abelian simple P groups are the P SL 2 (p) (p > 3 prime), and show how it follows that any P group G modulo its (odd) core is either a 2- group or isomorphic to some SL 2 (p) or T L 2 (p) (p 3 prime), where T L 2 (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 understanding 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

2 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 ]. 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 75 th birthday. 1 Notation and preliminaries We will use the following notation for (isomorphism classes of) groups: F n denotes the ring Z/nZ for any n N, and F n its multiplicative group of invertible elements. C n : cyclic of order n: x x n = 1. D 2n : dihedral of order 2n: x, y x n = y 2 = 1, y 1 xy = x 1. Q 4n : quaternionic of order 4n: x, y x n = y 2, y 4 = 1, y 1 xy = x 1. Groups Q(2 k l, m, n) are defined below. GL 2 (p): group of invertible 2 2 matrices over the Galois field F p ; throughout this paper, p will be an odd prime. SL 2 (p): matrices in GL 2 (p) with determinant 1. P GL 2 (p): quotient of GL 2 (p) by its centre (the group of scalar matrices). P SL 2 (p): the image of SL 2 (p) in P GL 2 (p). T L( 2 (p): let) ζ be the outer automorphism of SL 2 (p) induced by conjugation by ω 0 w :=, where ω generates F 0 1 p. Then T L 2 (p) = ( ) SL 2 (p), z z 1 gz = g ζ for all g SL 2 (p), z 2 ω 0 = 0 ω 1. (1) The four latter groups are all P groups and form a diagram SL 2 (p) T L 2 (p) P SL 2 (p) P GL 2 (p), (2) 2

3 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 = SL 2 (3), O = T L 2 (3) and I = SL 2 (5) for the binary tetrahedral, 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 G p 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. For p prime, write ν p (n) for the largest integer r with p r n. For r, n N, write ord n (r) for the order of r in F n. 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 homomorphism 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 corresponding to (A, B, h) iff a certain obstruction in H 3 (B; Z(A)) vanishes; these are then classified by H 2 (B; Z(A)). If A is abelian, then Z(A) = A and the zero element of H 2 (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 H 1 (B; A). Recall that if B and A have coprime orders, all groups H r (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 SL 2 (p) and T L 2 (p) is the group {±I} of order 2; the groups P SL 2 (p) and P GL 2 (p) have trivial centre. Lemma 1.1 The groups Out(P SL 2 (p)) and Out(SL 2 (p)) have order 2, all automorphisms being induced by P GL 2 (p), resp. GL 2 (p). ( ) ( ) Proof Write u := and t :=, both lying in SL (p). Every element of order p in GL 2 (p) is conjugate to u, but in SL 2 (p) these fall into two conjugacy classes. Let α be an automorphism of SL 2 (p) leaving u fixed. Then α preserves the normaliser of U := u, the group of upper triangular matrices, which is a semi-direct product of U and the group SD of diagonal matrices in SL 2 (p). Since the orders of these two are coprime, the splitting is unique up to conjugacy, so adjusting α by an inner automorphism, we may suppose that α preserves SD. Hence it also preserves ( the) normaliser of SD, which is an extension of D by t. 0 a Write α(t) =. Since this lies in SL b 0 2 (p), ab = 1. Since the relation (tu) 3 = I is preserved under α, we find a = 1 and b = 1, thus α(t) = t. Since SL 2 (p) is generated by t, u and SD, if α fixes SD it is the identity. Now α induces an automorphism α of P SL 2 (p). This fixes the images t, u of t, u and respects the image SD of SD: since SD acts faithfully on U, α fixes SD pointwise. Hence α is the identity. Thus for each g SL 2 (p) we have α(g) = ɛ(g)g, with ɛ(g) = ±I. Since α is a homomorphism and ±I is central, ɛ is a homomorphism, hence is trivial. 3

4 This proves the result for SL 2 (p); that for P SL 2 (p) follows easily. Proposition 1.2 (i) Any extension of P SL 2 (p), SL 2 (p), P GL 2 (p) or T L 2 (p) by a group of odd order splits as a direct product. (ii) No extension of P GL 2 (p) or T L 2 (p) by C 2 is a P group. (iii) Any extension of P SL 2 (p) resp. SL 2 (p) by C 2 which is a P group is isomorphic to P GL 2 (p) resp. T L 2 (p). Proof (i) In this case, h is trivial and the cohomology groups vanish (in this case, the orders of Z(A) and B are coprime): the result follows. (ii) For P GL 2 (p), there are no outer automorphisms, so h is trivial. The induced extension on a Sylow subgroup now cannot be dihedral. And if T L 2 (p) had an appropriate extension, we could factor out the centre and find one for P GL 2 (p). (iii) There are just two isomorphism classes of extensions of P SL 2 (p) by C 2, according as h : C 2 Out(P SL 2 (p)) is trivial or not, so any extension is either trivial or isomorphic to P GL 2 (p). There are four classes of extensions of SL 2 (p) by C 2 : h : C 2 Out(SL 2 (p)) may be trivial or not; in either case, C 2 acts trivially on the centre C 2 and H 2 (C 2 ; C 2 ) has order 2. When h is trivial, the corresponding extensions have Sylow subgroup containing a direct product of two cyclic groups, so are not P groups. For the others, the outer automorphism of SL 2 (p) was denoted ζ above, and we must adjoin to SL 2 (p) an element z which ( induces) ζ and has z 2 equal to an ω 0 element of SL 2 (p) inducing ζ 2, hence to ± 0 ω 1. Taking the + sign gives T L 2 (p), with quaternionic Sylow 2-subgroups; the minus sign gives a group whose Sylow 2-subgroups are semi-dihedral. We turn now to extensions of cyclic groups by the groups of (2). Since the automorphism group of a cyclic group is abelian, h factors through the commutator quotient group, which is trivial for P SL 2 (p) and SL 2 (p), except if p = 3, when it has order 3, and has order 2 for P GL 2 (p) and T L 2 (p). Proposition 1.3 (i) Any extension by P SL 2 (p) (p 3), SL 2 (p) (p 3), P GL 2 (p) or T L 2 (p) of a cyclic group of odd order is split. (ii) An extension of C 2 by P SL 2 (p) or P GL 2 (p) which is a P group is isomorphic to SL 2 (p) or T L 2 (p) respectively. (iii) No extension of C 2 by SL 2 (p) or T L 2 (p) is a P group. Proof (i) We must show that H 2 (G; C l ) vanishes for G one of the above and l odd; we may suppose l prime. The result is clear if l is prime to G. Otherwise, H 2 (G l ; C l ) = Hom(G l ; C l ) = C l, and the normaliser of G l in SL 2 (p) acts nontrivially on G l, so has no invariants in H 2 (G l ; C l ). (ii) Since Out(C 2 ) is trivial, extensions of C 2 by G are classified by H 2 (G; C 2 ). If G has Sylow 2-subgroup G 2, this maps injectively to H 2 (G 2 ; C 2 ), so the extension of G 2 is enough to determine that of G. First suppose G = P SL 2 (p). Then G 2 is dihedral, so has a presentation of the form x, y x 2k = y 2 = (xy) 2 = 1. An extension of C 2 by G 2 is thus of the form x, y, z z 2 = 1, xz = zx, yz = zy, x 2k = z a, y 2 = z b, (xy) 2 = z c, where the parameters a, b, c {0, 1} determine the different extensions. But the involutions x 2k 1, y, xy in P SL 2 (p) are all conjugate, so the induced extensions of C 2 by them are all isomorphic. Hence a = b = c. If a = 0, we have the trivial extension, which is not a P group. Thus a = 1, there is a unique extension, and this must be SL 2 (p). 4

5 An extension of C 2 by P GL 2 (p) has a subgroup of index 2 which is an extension by P SL 2 (p). If it is a P group, so is this subgroup, which must be isomorphic to SL 2 (p) by the above. By (iii) of Proposition 1.2, our group is isomorphic to T L 2 (p). This proves (ii). For (iii) we note that any extension of C 2 by SL 2 (p) is also an extension by P SL 2 (p) of a group T of order 4. If it is a P group, by Lemma 2.4, T is cyclic. Now (as above) a Sylow 2-subgroup of G has the form x, y, z z 4 = 1, xz = zx, yz = zy, x 2k = z a, y 2 = z b, (xy) 2 = z c, on noting that T must be central since any map from P SL 2 (p) to Aut(T ) is trivial. Since the quotient by z 2 is as in (ii) above, a, b and c are odd. But z a = y 1 z a y = y 1 x 2k y = (y 1 xy) 2k = (z b x 1 z c ) 2k = z 2k (c b) x 2k = z 2k (c b) a, so 2 k (c b) 2a is divisible by 4, the order of z. As k 1 and a, b and c are odd, we have a contradiction. Any extension of C 2 by T L 2 (p) has a subgroup of index 2 which is an extension of C 2 by SL 2 (p): since the latter cannot be a P group, nor can the former. 2 Structure of P groups Theorem 2.1 Any group whose Sylow subgroups are all cyclic is soluble. This result was proved by Burnside [9] over a century ago. A modern account of the proof is given in [47, 5.4.3]. The key result for us is the following, due to Suzuki [37]: Theorem 2.2 Let G be a simple group whose odd Sylow subgroups are cyclic and whose even ones are (i) dihedral or (ii) quaternionic. Then in case (i), G is isomorphic to P SL 2 (p) for some odd prime p > 3. Case (ii) cannot occur. The proof of (i) is contained in [37, 4,5]. No simpler proof is known, though the Brauer-Suzuki-Wall theorem [5], which is an essential tool, has been considerably simplified by Bender [1]. The result is, of course, contained in the Gorenstein-Walter classification [16] of groups with dihedral 2-subgroup, but this even in the much reduced version of Bender [2] can hardly be considered a simplification. The proof of (ii) is given in 6 of Suzuki s paper. It also follows at once from (i) and the following. Theorem 2.3 Let G be a finite group with a quaternionic Sylow 2-subgroup S. Then G has a normal subgroup Z such that Z is twice an odd number. The proof is given in the case S > 8 in [15, 12] and in case S = 8 in [14]. These arguments do not use block theory, so are technically simpler than Suzuki s original proof. For the above critique I am indebted to George Glauberman. Suzuki himself deduced that any non-solvable P group contains a normal subgroup G 1 = Z L of index 2, where Z is a (solvable) group whose Sylow subgroups are all cyclic, and L is isomorphic to P GL 2 (p) or SL 2 (p) for some p 5. To obtain a more precise result, we follow the method of Zassenhaus [48], and analyse the structure of a general P group by induction. First observe Lemma 2.4 Suppose G a P group, N normal in G. Then either N or G/N has cyclic Sylow 2 subgroups. Proof A Sylow 2-subgroup T of N lies in some Sylow 2-subgroup S of G. Then T = S N is normal in S, and S/T = NS/N is a Sylow 2-subgroup of G/N. If C is cyclic of index 2 in S, either T C is cyclic or T.C = S, so S/T = C/(C T ) is cyclic. 5

6 Theorem 2.5 If G is a P group, then G/O(G) is either a 2-group or isomorphic to P SL 2 (p), SL 2 (p), P GL 2 (p) or T L 2 (p) for some odd prime p. Proof By induction, we may suppose that G has a normal subgroup H satisfying the conclusion and such that G/H is a simple P group. Since H is normal in G and O(H) is characteristic in H, O(H) is normal in G. Factoring out O(H), we may now suppose O(H) = 1. First consider the case when H is isomorphic to P SL 2 (p), SL 2 (p), P GL 2 (p) or T L 2 (p). By Lemma 2.4, G/H has cyclic Sylow 2-subgroup; as it is simple, by Theorem 2.1 it is cyclic of prime order. If this order is odd, then by Proposition 1.2(i), the extension is trivial: G = H C, C = O(G) and G/O(G) = H. If it is even, G/H = C 2. By Proposition 1.2(ii), if H = P SL 2 (p) or SL 2 (p), then G = P GL 2 (p) or T L 2 (p). By Proposition 1.2(iii), we cannot have H = P GL 2 (p) or T L 2 (p) in this case. In the remaining case, H is a 2-group, and G/H is simple. According to Theorem 2.2, either G/H is cyclic of prime order or G/H = P SL 2 (p) with p 5. In the latter case, by Lemma 2.4, H has cyclic Sylow 2-subgroup, so is cyclic. If H = 2, it follows from Proposition 1.3(ii) that G = SL 2 (p). If H = 4, it follows from the same result that G is an extension of C 2 by SL 2 (p): by Proposition 1.3(iii), this cannot occur. The case H > 4 is likewise excluded by considering the quotient of G by a subgroup of index 4 in H. Otherwise, H is a 2-group and G/H is cyclic of prime order p. If p = 2, G is a 2-group. If p is odd, the extension is split, hence trivial (so G/O(G) = H) except when H has an automorphism of odd order, which occurs only when H is the four group or quaternion of order 8. Here G/H must have order 3, and the corresponding G is isomorphic to P SL 2 (3) resp. SL 2 (3). Corollary 2.6 For any P group G, G/O(G) is either a 2-group or isomorphic to SL 2 (p) or T L 2 (p) for some odd prime p. For as O(G) has odd order, the Sylow 2-subgroups of G/O(G) are isomorphic to those of G, hence are cyclic or quaternionic. Following Wolf [47], we define the type of G in terms of G/O(G) as follows G/O(G) C 2 k Q 2 k SL 2 (3) T L 2 (3) SL 2 (p) T L 2 (p) T ype I II III IV V V I, where V, V I are the non-soluble cases p > 3. 3 Presentations of P groups The group G has type I iff all its Sylow subgroups are cyclic. The structure of such groups was elucidated by Burnside [8]. A careful treatment is given by Wolf [47, 5.4.1] following Burnside and Zassenhaus [48]. Lemma 3.1 [47, 5.4.5] If G has cyclic Sylow subgroups, its commutator subgroup G is abelian, and its order is prime to its index. It follows that there is a presentation G = u, v u m = v n = 1, v 1 uv = u r. (3) For consistency we need n prime to m and r n 1 (mod m). Since G = u, (r 1) is prime to m. Set d := ord m (r): thus d n. 6

7 Any automorphism of G must be given by ψ a,b,c (u) = u a and ψ a,b,c (v) = v b u c for some a, b, c. For consistency we need (a, m) = 1, (b, n) = 1 and b 1 (mod d); conversely [47, 5.5.6(ii)] these conditions suffice. Since no power of u commutes with v, the map from G to Inn(G) is injective; its image consists of the ψ 1,1,c, while inner automorphism by v is ψ r,1,0. Decompose the cyclic group v of order n as a product of groups v 1, v 2 of coprime orders n 1, n 2, where each prime factor of n 1 divides d but prime factors of n 2 do not. Thus v 2 belongs to the centre of G and U := u, v 2 is cyclic. Any automorphism is given by (u, v 1, v 2 ) (u a, v b1 1 uc, v b2 2 ) with (a, m) = 1, (b 1, n 1 ) = 1, (b 2, n 2 ) = 1 and b 1 1 (mod d); and the condition b 1 1 (mod d) implies (b 1, n 1 ) = 1. Hence Aut(G) = φ(m). n1 d.m.φ(n 2). Since Z(G) = n/d, we have Inn(G) = md, so Out(G) = φ(m) d. n1 d.φ(n 2). The above is not the only way to present G as metacyclic. We will find it more convenient to use presentations of the form (3) where u generates the subgroup U. Thus the condition that (r 1) is prime to m is replaced by the condition that every prime factor of n divides d. This determines the subgroup u uniquely. We refer to these presentations as standard. Lemma 3.2 If G, of type I, has a standard presentation and H is prime to G, then any homomorphism h : H Out(G) has a unique lift to a homomorphism h : H Aut(G) such that, for all x H, h(v) = v. Proof Since U is cyclic, Aut(U) is abelian: write it as X Y, where X is prime to G and every prime divisor of Y divides G. Since U is characteristic in G, we have natural surjections Aut(G) Aut(U) X, whose composite factors through Out(G). Since the kernel has order prime to X, the surjection splits, and the splitting map is unique to conjugacy. By inspection, we see it can be taken to fix v. We next sharpen our description of Types III, IV. Following Wolf [47], we introduce the groups Tv := x, y, w x 4 = 1, y 2 = x 2, y 1 xy = x 1, w 3v = 1, w 1 xw = y, w 1 yw = xy, (4) Ov := Tv, z z 2 = x, z 1 yz = x 1 y, z 1 wz = w 1 x. (5) Thus T v is the split extension of the quaternion group Q 8 by C 3 v acting non-trivially. T v has centre C 3 v 1 generated by w 3, and the quotient by this is T 1 = T = SL 2 (3), the binary tetrahedral group. Also, O 1 = O. We now treat the case omitted in Proposition 1.3(i). Lemma 3.3 A P group which is an extension of C 3 v 1 to Tv. by SL 2 (3) is isomorphic Proof The induced extension by the Sylow 2-subgroup Q 8 of SL 2 (3) is trivial (coprime orders) so Q 8 is normal in G. The quotient is isomorphic to a Sylow 3- subgroup, which is cyclic since we have a P group. The extension of Q 8 by C 3 v is split, and the action of C 3 v on Q 8 is determined since it factors through the non-trivial action of C 3. Lemma 3.4 Any extension G of Tv Ov. by C 2 which is a P group is isomorphic to 7

8 Proof The extension must have Sylow subgroup Q 16, which is not a central extension of Q 8 by C 2. Thus the quotient G of G by Q 8 maps onto the outer automorphism group (dihedral of order 6) of Q 8, thus admits a presentation w, z w 3v = z 2 = 1, z 1 wz = w 1. This gives the structure of G and the map G Out(Q 8 ): since Z(Q 8 ) = C 2, the extension is determined by a class in H 2 (G; C 2 ), hence in turn by the extension of the Sylow 2-subgroup of G. To find a presentation, we may suppose z 2 = x; then z has order 8, so to have a quaternion group we need y 1 zy = z 1, so z 1 yz = x 1 y. Now z 1 wz = w 1 q for some q Q 8, and since conjugation by z must induce an automorphism of Tv we find we must have q = x. Theorem 3.5 Any P group G is an extension of a group G 0 of odd order by a group G 1 isomorphic to one of C 2 k (k 0), Q 2 k (k 3), T v or O v (v 1), SL 2 (p) or T L 2 (p) (p a prime 5). Moreover, we may suppose the orders of G 0 and G 1 coprime. Proof If G has type I or II, this follows from the definition of the type, with G 0 = O(G). If G has type V or VI, we again take G 0 = O(G). By Proposition 1.3, any extension by SL 2 (p) (p 3) or T L 2 (p) of a cyclic group of odd order is split. Now if l is a prime (necessarily odd) dividing both G 0 and G/G 0, a Sylow l subgroup of G is a split extension of non-trivial l groups, hence is not cyclic, contradicting our hypothesis. If G has type III, it is an extension by SL 2 (3) of O(G). As 3 is the smallest prime dividing O(G) it follows from [9, Art.128] that O(G) has a normal 3-complement G 0. Applying Lemma 3.3 to G/G 0, we can identify it with Tv for some v 1. By construction of G 0, its order is prime to 2 and 3, hence to G/G 0. If G has type IV, it has a subgroup of index 2 of type III, which by the above, we can write as an extension of G 0 by Tv. By Lemma 3.4, G/G 0 = O v. Since the extension given by Theorem 3.5 has coprime orders, it is split, and hence classified by h : G 1 Out(G 0 ). Now Out(G 0 ) is abelian, so G 1 maps via its commutator quotient, which is: C 2 k (type I), C 2 C 2 (type II), C 1 3 v for type III, C 2 for types IV, VI, and 1 for type V. We can now write down an explicit presentation for each type. For type I, this was already given in (3). For the rest, we already have a presentation of G 1, and take a standard presentation of G 0 (which has type I). By Lemma 3.2, the map G 1 Out(G 0 ) factors through the automorphism group of u. Hence we have Theorem 3.6 A P group G has a presentation as follows. If G has type I, use (3). Otherwise, take a standard presentation of the subgroup G 0, and a presentation of G 1 given by x, z x 2k 1 = 1, z 2 = x 2k 2, z 1 xz = x 1 for type II, (4) for type III, (5) for type IV, and (1) for type VI. Then add relations that all the generators of G 1 commute with v, and commute with u except as follows: II: x 1 ux = u a, z 1 uz = u b with a 2 b 2 1 (mod m), III: w 1 uw = u a with a 3v 1 (mod m), IV: z 1 uz = u a with a 2 1 (mod m), V: no exceptions, VI: z 1 uz = u a with a 2 1 (mod m). Corollary 3.7 A non-cyclic P group G is p hyperelementary if and only if either G has type I and n is a power of p or p = 2, G has type II, and n = 1. This follows by inspection. 1 incorrectly given in [41] as 3 8

9 4 Subgroups and refinement of type classification Suppose G is 2-hyperelementary of type II, hence an extension of a cyclic group G 0 of odd order by a quaternionic 2-group G 1 = Q. The extension is determined by the action of Q on G 0 ; since Aut(G 0 ) is abelian, Q acts through its commutator quotient group, which is a four group. As G 0 has odd order, so has unique square roots, any involution of G 0 determines a direct sum splitting, where the involution fixes one summand and inverts the other. A second involution commuting with the first preserves each summand and induce a further splitting of each. We thus have a direct product splitting G 0 = U 1 U i U j U k, (6) where x centralises U 1 and U i and inverts U j and U k, and z centralises U 1 and U j and inverts U i and U k. The notation is intended to suggest a representation Q 8 S 3 with x i, z j. Write m := U for = 1, i, j, k. Here the m i are odd and mutually coprime. In the notation of Milnor [33], this is denoted C m1 Q(2 k m i, m j, m k ). We note with Milnor that replacing y by xy will interchange the roles of m j and m k ; and that if Q has order 8, we can permute all of m i, m j and m k. For a general group G of type II, we use the presentation of Theorem 3.6 and apply the above splitting to the subgroup u, x, z to define parameters m. We will need information about hyperelementary subgroups of P groups. The next result will be the key to this. Theorem 4.1 For G of one of the types III-VI, any subgroup H of G of type I or II is, up to conjugacy, contained in one of the pre-images H, H in G of the subgroups H 1, H 1 of G 1 = G/G0 indicated in the following table: G 1 H 1 H 1 SL 2 (p) Q 2(p±1) C p.c p 1 T L 2 (p) Q 4(p±1) C p.c 2(p 1) Tv Q 8 C 3 v 1 C 2.3 v Ov Q(16, 3 v 1, 1) Q 4.3 v Proof If H is a subgroup of G of type I or II, its image in G/G 0 = G1 has the same type, so is contained in a maximal such subgroup H 1. The pre-image of H 1 in G has the same type as H 1. Thus it suffices to consider subgroups of G 1. Since a group of type I or II normalises a non-trivial cyclic subgroup, it suffices to list such normalisers in G 1. Define, for g GL 2 (p), M(g) := {x GL 2 (p) : x 1 gx = λg r for some λ F p, r Z}, with image P M(g) in P GL 2 (p), and SM(g) := M(g) SL 2 (p). If g has distinct eigenvalues in F p, M(g) is a wreath product C 2 F p, SM(g) = Q 2(p 1) is quaternionic and P M(g) = D 2(p 1) is dihedral. The case when the eigenvalues of g do not lie in F p is dealt with by considering subgroups of GL 2 (F p 2) and then taking invariants under the Galois group. We find that SM(g) = Q 2(p+1) is quaternionic and P M(g) = D 2(p+1). If g is not semi-simple but not central, M(g) is conjugate to the group of upper triangular matrices, SM(g) is the split extension of F + p = C p by F p = C p 1 with the square of the natural action, and P M(g) the corresponding extension with the natural action. The list of normalisers of cyclic subgroups of T L 2 (p) is obtained from the list for P GL 2 (p) by lifting. 9

10 Given H Tv, if the projection H onto the quotient group C 3 v is not surjective, H is a subgroup of x, y, w 3 Q 8 C 3 v 1. Otherwise H contains an element wq with q Q 8 : up to conjugacy, this is either w or wx 2, hence H w, x 2 C 2.3 v, which is a maximal proper subgroup. For H Ov, the intersection H Tv must be contained in one of the above. In the first case, H is contained in the pre-image of a subgroup of order 2 of Out(Q 8 ), conjugate to x, y, z, w 3 : x, y, z = Q 16 and w 3 is centralised by y and inverted by z. In the second, w must be normal in H; we check that ζ = zxy satisfies ζ 1 wζ = w 1 and ζ 2 = x 2 so ζ, w = Q 4.3 v. It was shown over a century ago by Dixon that this gives all maximal subgroups of the SL 2 (p) except for binary tetrahedral, octahedral and icosahedral groups. We next determine, for each G of type III-VI, presented as in Theorem 3.6, the structure of the groups H and H just defined. The presentation (3) of a group of type I is defined by parameters (m, n, r); we will list (m, n; d) where, as above, d = ord m (r). The presentation of a group of type II is defined by parameters (m, n, r, k, a, b); we again list d rather than r and list the quadruple (m 1, m i, m j, m k ) introduced in (6) rather than (m, a, b), so give (m 1, m i, m j, m k ; n, 2 k ; d). For SL 2 (p) and T L 2 (p), denote by H η (η = ±1) the cases for H corresponding to Q 2(p+η) and Q 4(p+η) ; also set ɛ = ±1 with p ɛ (mod 4); and write p ɛ = qp with q a power of 2 and p odd. For G 1 = Tv or Ov we need to distinguish according as the parameter a in the presentation takes the value 1 or not; in the former case, write 3 s = ord m (a). For types IV, VI we have a relation z 1 uz = u a and use this to split U (as in (6) with factors of orders m +, m. Theorem 4.2 The subgroups listed in Theorem 4.1 have the following invariants. The groups H all have type I, with parameters G 1 H 1 H (a 1) H (a = 1) SL 2 (p) C p.c p 1 (pm, (p 1)n; 1 2 (p 1)d) T L 2 (p) C p.c 2(p 1) (pm, 2(p 1)n; (p 1)d). Tv C 2.3 v (2m, 3 v n; 3 s d) (2.3 v m, n; d) Ov Q 4.3 v (3 v m, 4n; 2d) (4.3 v m, n; d) The groups H all have type II except H ɛ in the SL 2 (p) case, which has type I with parameters ( 1 2m(p + ɛ), 4n, 2d). Their parameters are given by G 1 H 1 m 1 m i m j m k n q d SL 2 (p) Q 2(p ɛ) m p 1 1 n 2q d T L 2 (p) Q 4(p ɛ) m + p m 1 n 4q d 1 T L 2 (p) Q 4(p+ɛ) m + 2 (p + ɛ) m 1 n 8 d Tv (s > 1) Q 8 C 3 v 1 m v 1 n 8 3 s 1 d Tv (s = 0, 1) Q 8 C 3 v 1 3 v 1 m n 8 d Ov Q(16, 3 v 1, 1) m v 1 m 1 n 16 d Proof The type of H is the same as that of H 1. In each case, H is an extension of G 0 by H 1, the two having co-prime orders. Recall that all generators of G 1 commute with those of G 0 except for: (type III) w 1 uw = u a, (types IV,VI) z 1 uz = u a. The results for H of type I now follow by inspection. For G of type III, H has Q 8 as a direct factor; the other factor has type I and invariants (m, 3 v 1 n, 3 s 1 d) if s > 1 or (3 v 1 m, n, d) if s = 0, 1. 10

11 In the remaining cases, no element of odd order in G 1 acts non-trivially on a cyclic subgroup of G 0, so the parameters n and d for H are the same as those for G 0. The parameter 2 k is the highest power of 2 dividing H. It remains only to find the m : recall that for C a Q 2k b with a, b odd, these are (a, b, 1, 1). For SL 2 (p) and H 1 = Q 2(p ɛ), it suffices to note that the quaternion subgroup centralises u. For the case T L 2 (p), however, it is the generator z of highest order which inverts the summand C m, giving the result in that case. In the final case, we had x, y, z = Q 16 and w 3 is centralised by y and inverted by z; the same holds for C m. For G of type II, with the notation of (6), we say that G is of rtype IIK if we can arrange that m j = m k = 1, rtype IIL if Q 16 and m j m k 1, rtype IIM if Q = 8 and two of m i, m j and m k are 1. Thus G has rtype IIK if and only if it has a cyclic subgroup of index 2. We will write IILM for a group whose rtype may be IIL or IIM. If G has type II, and is presented as in Theorem 3.6, the subgroup H := u, x, y is 2-hyperelementary and has the same parameters m, hence the same rtype, as G. Proposition 4.3 For G of type II, every subgroup H of G of type II has rtype IIK or has the same rtype as G. Proof We may replace H by a 2-hyperelementary subgroup of the same rtype, and so suppose H 2-hyperelementary. Then H normalises a cyclic subgroup Z of odd order. Replacing Z by a conjugate, we may suppose Z G 0, hence that Z is the direct product of a subgroup of u and a subgroup of v. The Sylow 2- subgroup S of H is conjugate in the normaliser of Z to a subgroup of x, z, so may be supposed a subgroup of this. Now if S = x, z, the parameters m (H) m (G), so the result follows. Otherwise S x 2, z (or x 2, xz ), and since x 2 centralises Z, H has type IIK in this case. Theorem 4.4 (i) If G has type III or V, G has no subgroup of rtype IIL or IIM. (ii) If G has type VI, then either (a) z commutes with u, G = G 0 T L 2 (p) is a product (coprime orders), and every type II subgroup has rtype IIK, or (b) z and u do not commute and G has subgroups of rtypes IIK, IIL and IIM. (iii) If G has type IV, then either (a) v = 1, z commutes with u, G = G 0 O is a product (coprime orders), and every type II subgroup has rtype IIK, or (b) G has subgroups of rtypes IIK and IIL, but none of rtype IIM. Proof By Theorem 4.1, every hyperelementary subgroup H of G is contained in one of the subgroups H, H. These are described in Theorem 4.2; in particular, H has type I. It follows from Proposition 4.3 that if G has a subgroup of rtype IIL or IIM, then one of the subgroups H has that rtype. Now (i) follows by inspection of the list. If G has type VI, then if m = 1 we only obtain type IIK; if m 1, the pre-image of Q 4(p ɛ) has rtype IIL; the pre-image of Q 4(p+ɛ) has rtype IIM. This implies (ii). If G has type IV, then if 3 v 1 m = 1 we only obtain type IIK, otherwise H has type IIL. Hence (iii) holds. We define G to have rtype VIK resp. VIL in cases (iia), (iib) in the above Theorem, and to have rtype IVK resp. IVL in cases (iiia), (iiib). 11

12 The p period of a finite group G is the period (if any) for the p primary part of its cohomology; we denote it by 2P p (G): equivalently, 2P p (G) is the least period for projective resolutions of Ẑp(G). Thus for G a P group, its cohomology has period 2P (G) with P (G) the least common multiple of the P p (G). We compare this with the Artin exponent e(g) [22], the least positive integer such that e(g)1 belongs to the ideal of the rational representation ring generated by representations induced from cyclic subgroups. Lemma 4.5 (di) P 2 (G) = 1 (resp. 2) if the Sylow 2-subgroup of G is cyclic resp. quaternionic. (dii) If p is odd, and G p is a cyclic Sylow p subgroup of G with normaliser N p, then P p (G) equals the order of the group of automorphisms induced on G p by N p. (diii) For a P group G, the period 2P (G) is the least common multiple of the periods of hyperelementary subgroups. (ei) If H is a hyperelementary P group of type I, we have e(h) = P (H). (eii) If H is a 2-hyperelementary P group, we have e(h) = 2 for H of rtype IIK, e(h) = 4 for H of rtype IIL or IIM. (eiii) For any G, e(g) is the least common multiple of the e(h), H a hyperelementary subgroup of G. Here (di)-(diii) are due to Swan [39], (ei)-(eiii) to Lam [22]. We now calculate P (G) and e(g) in terms of the notation of Theorem 3.6. We see at once that for G of type I, we have e(g) = P (G) = d; also that for type IIK, e(g) = P (G) = 2d, while for type IILM we have P (G) = 2d, e(g) = 4d. By (diii) (resp. (eiii)), P (G) (resp. e(g)) is the least common multiple of the P (H) (resp. e(h)) for H a subgroup of G of type I or II, and so it follows from Theorem 4.1 that for G of type III-VI it suffices to consider the subgroups there listed. Their parameters are given in Theorem 4.2, from which the results can be read off. T ype III IV K IV L V V IK V IL P (H ) = e(h ) 3 s 1 d d 2d 2 (p 1)d (p 1)d (p 1)d P (H ±1) 2.3 s 1 d 2d 2d 2d 2d 2d e(h ±1) 2.3 s 1 d 2d 4d 2d 2d 4d (7) Corollary 4.6 The period of a P group G is given by T ype of G I II III IV V (ɛ = 1) V (ɛ = 1) V I P (G) d 2d 2.3 s 1.d 2d 2 (p 1)d (p 1)d (p 1)d For G a P group, e(g) = P (G) except when ν 2 (P (G)) = 1 and G has a subgroup of rtype IIL or IIM; equivalently, G has rtype IIL, IIM, IVL or VIL with ɛ = 1; and in these cases, e(g) = 2P (G). 5 Free orthogonal actions We call a representation ρ : G U N of G an F representation if the induced action of G on the unit sphere S 2N 1 C N is free. We next give Wolf s [47] classification of F representations. For p, q primes (not necessarily distinct), one says that G satisfies the pq condition if every subgroup of G of order pq is cyclic. Thus G is a P group if and only if it satisfies all p 2 conditions. Theorem 5.1 [47, , 6.3.1]. The following are equivalent: (i) G has an F representation, 12

13 (ii) G satisfies all pq conditions and has no subgroup P SL 2 (p) with p > 5, (iii) G is a P group, such that in the notation of Theorem 3.6, n/d is divisible by every prime divisor of d, and G 1 is not SL 2 (p) or T L 2 (p) with p > 5. To show that (i) implies (ii) it suffices to verify that a non-cyclic group of order pq, or a group SL 2 (p) with p > 5, has no F representation: it is elementary to construct all irreducible representations ρ and check that in each case, for some element g 1, ρ(g) has 1 as an eigenvalue. To see that (ii) implies (iii) observe first, that since G satisfies all p 2 conditions, it is a P group, so can be put in our normal form and second, that if p d but p n, there is some prime q m on which the action of C p is non-trivial, so G has a non-cyclic subgroup of order pq. That (iii) implies (i) follows from the explicit construction of F representations, which we give next, again following Wolf. Lemma 5.2 (i) If G is cyclic, the irreducible F representations are the faithful 1-dimensional representations. (ii) T has a unique irreducible F representation; for v > 1, Tv has 2.3 v 1 irreducible F representations; all have degree 2. (iii) O has 2 irreducible F representations, both of degree 2. (iv) I has 2 irreducible F representations, both of degree 2. (v) The irreducible F representations of a direct product of groups of coprime orders are the (external) tensor products of those of the factors. Here (i) is elementary, the rest obtained by standard representation theory in [47, 7.1.3, 7.1.5, 7.1.7, 6.3.2] respectively. In (iii) and (iv), the two representations are equivalent under the outer automorphism of G; the images of T, O, I are, of course, the binary polyhedral groups. In the general case, the result can be stated as follows. Theorem 5.3 The irreducible F representations of G are induced from those of the subgroup R(G) defined as follows. In each case write R 0 for the cyclic subgroup u, v d. If G has type I, take R(G) = R 0. If G has type III, write R 1 := Ker(G 1 Out(G 0 )), so R 1 = Tv if s = 0 and R 1 = Q8 C 3 v s if s > 0. Take R(G) = R 0 R 1. If G has type IVK or V, take R(G) = R 0 G 1. If G has type II, IVL or VI, the subgroup G + of index 2 defined by omitting z from the list of generators has type I, III or V respectively. Take R(G) = R(G + ). The proof occupies [47, 7.2]; the result is tabulated in Wolf s (very different) notation in [47, ]. In each case, R(G) G; the irreducible F representations of R(G) are given by Lemma 5.2 and have degree δ = 1 if G has type I or II, and 2 otherwise. Note that if G has type IILM with parameters (m 1, m i, m j, m k ; n, 2 k, d), G + has parameters (m 1 m i m j m k, 2 k 1 n; 2d). The quotient G/R(G) is cyclic except if G has type IILM, and has order rtype of G I IIK IILM III IV K IV L V V I G : R(G) d 2d 4d 3 s.d d 2d d 2d It follows that the irreducible F representations of G all have the same degree δ G : R(G). On comparing with Corollary 4.6, we see that, in each case, this degree equals e(g) (since p = 5 here, the case p 1 (mod 4) does not arise). To see that the induced representations are indeed fixed-point free, it is necessary to check that, for each g G \ R(G) with class of order b, say, in G/R(G), we 13

14 have g b 1: it suffices to consider the cases where b is prime. Each F representation ρ of G (of degree N) defines a free action of G on S 2N 1, and hence a quotient manifold X ρ := S 2N 1 /G. The first Postnikov invariant of X ρ in H 2N (G; Z) can be identified with the N th Chern class c N (ρ). We calculated these classes for G hyperelementary in [46, Theorem 11.1]: as the calculation is delicate, we now repeat it for the somewhat more general case of all groups of type I or II. We first consider the eigenvalues of each ρ(g). Given representations ρ of G and σ of H, if ρ(g) has eigenvalues α i and σ(h) has eigenvalues β j, then the eigenvalues of the external tensor product (ρ σ)(g, h) are the α i β j. Now suppose that H G has quotient C d and ρ is a representation of H. Choose a left transversal {g i } for H in G (i.e. G = i Hg i). Then for g G, the eigenvalues of Ind G Hρ(g) are: if g H, the union of the sets of eigenvalues of the ρ(g 1 i gg i ); if g H, the d th roots of the eigenvalues of ρ(g d ). In particular, as noted above, Ind G Hρ is a F representation if and only if ρ is an F representation and g d 1 for g H. We need a precise notation for cohomology classes. Consider the cyclic group G = C m generated by x. There is a natural isomorphism i G : Hom(G, S 1 ) H 2 (G; Z), the boundary map in the exact sequence H 1 (G; R) H 1 (G; S 1 ) H 2 (G; Z) H 2 (G; R), whose extreme terms vanish. For each character ρ, i G (ρ) is the Chern class c 1 (ρ), and is a cohomology generator. The irreducible F representations of G have dimension 1, and are given by ρ k (x) = e 2πik/m with k prime to m. Set ˆx := i G (ρ 1 ) = c 1 (ρ 1 ): then c 1 (ρ k ) = kˆx. A direct sum ρ := t i=1 ρ k b of representations gives an action of G on the join of the corresponding spheres, and c t (ρ) = i c 1(ρ kb ) = ( i k b)ˆx t. It is important to note that here we have k b, not k b, which is what appears in det(ρ(x)) = e 2πi k b /m. Next let G = Q 8n be a quaternionic 2-group, with presentation x, y x 4n = 1, y 2 = x 2n, y 1 xy = x 1, and cyclic subgroup G + = x of index 2. The irreducible F representations of G are induced from those of G +, which are the ρ r with r odd: set σ r := Ind G G ρ + r. The restriction of σ r to G + is ρ r ρ r, so has Chern class c 2 (σ r ) = r 2ˆx 2. Up to isomorphism, there are 2n different ρ r and n different σ r. We define ˆX H 4 (G; Z) to be c 2 (σ 1 ). Then we have c 2 (σ r ) = r 2 ˆX; this is less trivial than the corresponding result in the cyclic case: one proof was given in [46, 11.2]; we can also write down an equivariant map of degree r 2 between the representation spaces. Note that H 4 (G; Z) is cyclic of order 8n: the value of r 2 modulo 8n is determined by that of r modulo 4n, and is 1 (mod 8); the value of c 2 (σ r ) determines σ r up to isomorphism. Theorem 5.4 Let G be of type I, admitting F representations; use the above notation. Then the irreducible F representations σ s,t of G have degree d and their Chern classes c d are the cohomology generators which restrict to each G l as d th powers multiplied by λ l (G), where for l m, λ l (G) = 1 for d odd, λ l (G) = r d/2 for d even, for l n, λ l (G) = 1 for l or d odd; if ν 2 (n/d) 2, λ 2 (G) = 1+n/2; if ν 2 (n/d) = 1 and ν 2 (n) 3, λ 2 (G) = 1; if ν 2 (d) = 1 and ν 2 (n) = 2, λ 2 (G) = 1. Proof We have G = u, v u m = v n = 1, v 1 uv = u r, with n prime to m and r n 1 (mod m), d = ord m (r) and each prime divisor of n divides both d and n/d. Then K := u, v d is cyclic, with F representations given by σ s,t (uv d ) = e 2πi(s/m+td/n) with s prime to m and t prime to n. The irreducible F representations of G are the π s,t = Ind G K σ s,t. The eigenvalues of π s,t (z) are as follows: 14

15 if z = u, the e 2πisrj /m, (0 j < d); if z = v d, e 2πitd/n (d times); if z = v, the e 2πi(t/n+j/d), (0 j < d). Thus c d (π s,t ), restricted to u, is d 1 j=0 (srj )û d = s d r 1 2 d(d 1) û d ; the restriction to v is d 1 j=0 (t + jn/d)ˆvd. First consider r 1 2 d(d 1) modulo m. Since r d 1 (mod m), if d is odd, 1 2d(d 1) is divisible by d and we have r 1 2 d(d 1) +1 (mod m). If d is even, we see similarly that r 1 2 d(d 1) = r 1 2 d (mod m). In the latter case, n is even, so m is odd. Secondly, we need d 1 j=0 (t + jn/d) (mod n). We argue as follows, after [46, p 538]. The residues (mod n) of the 1 + jn/d (0 j d 1) form a subgroup of F n, since d and n/d have the same prime divisors. Most elements of the subgroup cancel (mod n) with their inverses when we multiply, so we must find the elements of order 2. Now the group is the direct product of its primary subgroups, corresponding to prime factors p n, so elements of order 2 come from p = 2. If ν 2 (n/d) 2, the Sylow 2-subgroup is cyclic and the only element of order 2 is 1 +n/2; if ν 2 (n/d) = 1 and ν 2 (n) 3, we have the 4 elements ±1, ±(1 + n/2), whose product is 1 (mod n); if ν 2 (d) = 1 and ν 2 (n) = 2, we have the two elements ±1, with product -1. The conclusion can be re-stated as follows. For any G of type I, define λ l (G) = ( 1) P (G)/Pl(G) for l odd, λ 2 (G) = ν2( G ) 1 if ν 2 ( G ) 2 ν 2 (P (G)) 1 or (ν 2 ( G ), ν 2 (P (G))) = (2, 1), λ 2 (G) = 1 otherwise. (8) Corollary 5.5 The theorem also holds with this notation. Proof The factor r 1 2 d can only take the values ±1 (mod G l ), and takes the value 1 only if (l is odd and) d/ord Gl (r) is odd. But P (G) = d and by Lemma 4.5, P l (G) = ord Gl (r). If d is odd, 1 is a d th power, so the sign is irrelevant. If l n and d is even, P l (G) is 1, so ( 1) P (G)/Pl(G) = 1. For G of type II, by Theorem 5.1, G has F representations if and only if each prime divisor of n divides both d and n/d. We have the presentation of Theorem 3.6 but, as in 4, we split U := u as a direct product U 1 U i U j U k. For any group G of type II, define parameters for l odd, λ l (G) = 1 if G has rtype IIK and l m i and λ l (G) = +1 otherwise, λ 2 (G) = 1 for G of rtype IIK, λ 2 (G) = (1 + 2 k 2 ) 2 for G of rtype IILM. Recall that e(g) = 2d for G of rtype IIK and 4d for G of rtype IIKL. Theorem 5.6 Let G be a P group of type II admitting F representations. Then the irreducible F representations of G have degree e(g), and their top Chern classes are the cohomology generators which restrict to each G l (l odd) as e(g) th powers multiplied by λ l (G); and to G 2 as λ 2 (G) ˆX e(g)/2 multiplied by an e(g) th power. Proof It follows from Theorem 5.3 that the F representations of G are obtained as follows. The group G + := u, v, x has type I. If G has rtype IIK, U j = U k = {1}, and R(G + ) = u, v d, x ; for rtype IILM, R(G + ) = u, v d, x 2. The irreducible F representations of G + are induced from the irreducible F representations of R(G + ), and in turn induce the irreducible F representations of G, which thus have degree e(g), which is 2d, 4d in the two cases. For G of rtype IIK, the parameter d for G + is odd. Applying Theorem 5.4 to G +, we find that the Chern classes c d of the irreducible F representations ρ of G + 15

16 are the generators which restrict to l subgroups as d th powers. Let g G + have order l a with l odd. Write the eigenvalues of ρ(g) as e 2πik/la for k = k 1,..., k d, then if l m 1 n those of Ind G G (ρ)(g) are the same, each repeated; if l m + i, we must add the values k = k 1,..., k d. Multiplying up, since b k b is a d th power we obtain, in the first case a (2d) th power, and in the second case, the negative of one. For the Sylow 2-subgroup, inducing up σ kb leads to kb 2 ˆX, so we obtain ˆX d multiplied by a (2d) th power. If G has rtype IILM, the parameter d for the group G + is 2d in our notation, so is even. For g G + of order l a with l odd with the eigenvalues of ρ(g) {e 2πik/la k = k 1,..., k 2d }, then if l divides m 1 m i n those of Ind G G (ρ)(g) are the + same, each counted twice; if l m j m k, we must add the values k = k 1,..., k 2d, thus the product in each case is ( k b ) 2. Since b k b is λ l (G) times a (2d) th power and λ l (G) 2 1, in each case we obtain an arbitrary (4d) th power. For l = 2, each eigenvalue e 2πik b/2 k 2 of x 2 in the representation of R(G) gives eigenvalues ±e 2πik b/2 k 1 of x G + when we induce up. Since k b + 2 k 2 k b (1 + 2 k 2 ) (mod 2 k 1 ), the class is d b=1 (k2 ˆX.k b b 2(1 + 2k 2 ) 2 ˆX). Since d is odd and (1 + 2 k 2 ) 4 1 (mod 2 k ), we have (1 + 2 k 2 ) 2 ˆX4d multiplied by a (4d) th power. We can state the condition for G 2 more explicitly: the (2d) th powers modulo 2 k are the same as the squares, and give all numbers 1 (mod 8); the (4d) th powers give all numbers 1 (mod 16); and the class of (1 + 2 k 2 ) 2 (mod (4d) th powers) is 1 if k = 3, k 1 if k 4. 6 The finiteness obstruction A finitely dominated CW complex X with fundamental group G and universal cover homotopy equivalent to S N 1 is said to be a (G, N)-space. The first Postnikov invariant g H N (G; Z) of X has additive order G, and multiplication by g induces isomorphisms of cohomology groups of G in positive dimensions; if we use complete (Tate) cohomology (see [10, Chapter XII]), this holds in all dimensions. An element g with this property is called a (cohomology) generator. It is known (see [40] and [41, Theorem 2.2]) that homotopy types of oriented (G, N)-spaces X correspond bijectively to cohomology generators g. Denote by X g a (G, N)-space corresponding to g H N (G; Z); then X g is a Poincaré complex. The chain complex of the universal cover X g is chain homotopy equivalent to a sequence P of finitely generated projective ZG modules yielding an exact sequence 0 Z P N 1... P 1 P 0 Z 0. The class of this sequence in Ext N ZG (Z, Z) = H N (G; Z) is g. Swan s finiteness obstruction o(g) K 0 (ZG) is defined as the class of n 1 0 ( 1) i [P i ]: it was shown in [40] that it vanishes if and only if there is a finite CW-complex homotopy equivalent to X. We have o(g 1 g 2 ) = o(g 1 ) + o(g 2 ). (9) The generators g in complete cohomology form a multiplicative group Gen(G). By (9), we have a homomorphism o : Gen(G) K 0 (G). Taking degree gives a homomorphism Gen(G) Z with image 2P (G)Z. Its kernel, the torsion subgroup of Gen(G), is the group Ĥ0 (G; Z) = F M, where M = G. We denote the restriction by o 0 : F M K 0 (G), and its image, the so-called Swan subgroup, by Sw(G) K 0 (G). Lemma 6.1 (i) If r Z is prime to M, o 0 (g) is the class of r, I G ZG. (ii) o is natural for restriction to subgroups. (iii) The map K 0 (ZG) {K 0 (ZH) H G hyperelementary} is injective. 16