A COHOMOLOGICAL PROPERTY OF FINITE p-groups 1. INTRODUCTION

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1 A COHOMOLOGICAL PROPERTY OF FINITE p-groups PETER J. CAMERON AND THOMAS W. MÜLLER ABSTRACT. We define and study a certain cohomological property of finite p-groups to be of Frobenius type, which is implicit in Frobenius theorem Berl. Sitz. 1895, concerning the equation X m = 1 and Philip Hall s subsequent work Proc. London Math. Soc. 40, on equations in finite groups. Hall s twisted version loc. cit., Theorem 1.6 of Frobenius theorem implies that each cyclic group of prime power order is of Frobenius type. We show that this property in fact pertains to every finite p-group. 1. INTRODUCTION One of the most beautiful results of Philip Hall s seminal paper [4] provides a twisted version of Frobenius well-known theorem 1 concerning the equation X m = 1 in finite groups. Hall s original result [4, Theorem 1.6], which is expressed in terms of solution numbers of a certain type of equation over finite groups, can be restated, in terms more germane to the present investigation, as follows. Proposition. Let C be a finite cyclic group, H a finite group, and let α : C AutH be an action by automorphisms of C on H. Then Der α C,H 0 mod gcd C, H. Hall s theorem reduces to Frobenius result upon setting α = 1. Recall the concept of a derivation in a non-commutative setting: given groups G and H, and a fixed action α : G AutH by automorphisms of G on H, a map d : G H is called a derivation with respect to the action α, if dg 1 g 2 = dg 1 αg 2 dg2 g 1,g 2 G. Note that, for a derivation d : G H with respect to α, we have d1 = 1 and, consequently, dg 1 αg = dg 1 g G. 1 We denote by Der α G,H the set of all derivations d : G H with respect to a given action α : G AutH. Definition. Let p be a prime. i A non-trivial finite p-group G is termed admissible, if, for each finite group H with p H and every action α : G AutH, the corresponding set Derα G,H of derivations d : G H, formed with respect to this action α, has cardinality a multiple of p. ii A finite p-group G is said to be of Frobenius type, if every subgroup U > 1 of G is admissible. 1 See [1, 2, Theorem II] and [3]. 1

2 2 P. J. CAMERON AND T. W. MÜLLER In this terminology, Hall s result above immediately implies the following. Corollary. Every cyclic group of prime power order is of Frobenius type. The question addressed in the present note is: which finite groups of prime power order are of Frobenius type? This problem was raised in [6], having arisen implicitly in [5] in connection with a descent principle in modular subgroup arithmetic. The main result of this paper provides a somewhat surprising answer to this problem. Theorem. Every group of prime power order is of Frobenius type. The proof is contained in the next two sections. 2. THE UNTWISTED CASE In order to establish our theorem, we first have to deal with the untwisted case α = 1. Lemma. Let p be a prime, G a non-trivial finite p-group, and let H be a finite group of order divisible by p. Then HomG,H 0 mod p. Proof. Classifying homomorphisms by their kernel, and applying the isomorphism theorem, we find that HomG,H = InjG/V, H. 2 V G We now make use of the facts that i every subgroup of index p in a finite p-group is normal, ii the automorphism group of G/V contains an element of order p, provided G/V > p, and iii AutG/V acts freely on the set InjG/V,H, provided the latter is non-empty. The second fact follows, for instance, from Gaschütz s theorem [2] asserting the existence of an outer automorphism of order p; however, since we only need to know that p AutG/V for G/V > p, we can get by with a more elementary argument. Indeed, if G/V is non-abelian, then it has an inner automorphism of order p. If, on the other hand, G/V is abelian and G/V > p, then G/V must contain a direct summand of one of the forms C p σ or σc p with σ 2. In the first case, AutG/V is divisible by AutC p σ = ϕp σ = p σ 1 p 1 0 mod p, where ϕ is Euler s totient function, while, in the second case, AutG/V must be divisible by AutσC p = GL σ p = p σ 1p σ p p σ p σ 1 0 mod p. Hence, evaluating 2 modulo p, we get HomG,H 1 + s p G InjC p,h mod p. We have s p G 1 mod p by Frobenius generalization of Sylow s third theorem and the fact that G 1. Alternatively, one might compute s p G = prg 1 p 1 = 1 + p + p p rg 1 1 mod p,

3 A COHOMOLOGICAL PROPERTY OF FINITE p-groups 3 where rg is the rank of the factor group G = G/ΦG, since every subgroup of G of index p contains ΦG. Moreover, by Frobenius theorem concerning the equation X m = 1 in finite groups and the fact that p H, we have whence the lemma. InjC p,h 1 mod p, 3. PROOF OF THE THEOREM Let p be a prime, G a non-trivial finite p-group, H a finite group of order divisible by p, and let α : G AutH be an action by automorphisms of G on H, where multiplication in the group AutH is given by the rule We have to show that Let σ 1 σ 2 h := σ 2 σ 1 h σ 1,σ 2 AutH, h H. Der α G,H 0 mod p. 3 F G,H = H G be the set of all functions from G to H. We make G act from the right on F G,H by setting f gx := fgxg 1 αg g,x G, f F G,H. For g 1,g 2,x G and f F G,H, f g1 g 2 x = f g1 g 2 xg 1 2 αg 2 = fg 1 g 2 xg 1 2 g 1 1 αg 1 g 2 = f g 1 g 2 x, as well as f 1 = f, and we have indeed defined an action of G on F G,H. In what follows, two distinguished subsets of F G,H will play a role: the set Der α G,H of derivations d : G H with respect to α, and the set Hom G,H of anti-homomorphisms from G to H. Note that a bijection being given by the map HomG,H = Hom G,H, 4 ψ ψ, ψ g := ψg 1 g G, ψ HomG,H. We have to check that the action of G on F G,H defined above restricts to an action of G on the subsets Der α G,H and Hom G,H. Indeed, for g,x,y G and d Der α G,H, we have d gxy = dgxyg 1 αg = dgxg 1 αgy dgyg 1 αg = d gx αy d gy,

4 4 P. J. CAMERON AND T. W. MÜLLER that is, d g : G H is again a derivation with respect to α. Similarly, for g,x,y G and ψ Hom G,H, ψ gxy = ψ gxyg 1 αg = ψ gyg 1 αg ψ gxg 1 αg = ψ gyψ gx. Denote by Der α G,H G and Hom G,H G the fixed point sets of Der α G,H and Hom G,H, respectively, under the respective G-action, so that and The decisive point in the proof is the fact that Der α G,H Der α G,H G mod p 5 Hom G,H Hom G,H G mod p. 6 Der α G,H G = Der α G,H Hom G,H = Hom G,H G. 7 Indeed, let d Der α G,H. Then Now, using 1, d Der α G,H G dgxg 1 αg = dx g,x G. dgxg 1 αg dgx αg = 1 αg dg 1 = dgx dg 1 αg = dgx dg 1, hence d Der α G,H G d Hom G,H, d Der α G,H. Similarly, if ψ Hom G,H, then ψ Hom G,H G ψ g 1 xg = ψ x αg g,x G ψ xgψ g 1 = ψ x αg g,x G ψ Der α G,H, whence 7. In view of equations 4 7 and the lemma, we now find that, modulo p, Der α G,H Der α G,H G = Der α G,H Hom G,H = Hom G,H G Hom G,H = HomG,H 0, whence 3, and the proof of the theorem is complete.

5 A COHOMOLOGICAL PROPERTY OF FINITE p-groups 5 REFERENCES [1] G. Frobenius, Verallgemeinerung des Sylow schen Satzes, Berliner Sitzungsber. 1895, [2] W. Gaschütz, Nichtabelsche p-gruppen besitzen äußere p-automorphismen, J. Algebra , 1 2. [3] I. M. Isaacs and G. R. Robinson, On a theorem of Frobenius: solutions of x n = 1 in finite groups, Am. Math. Monthly , [4] P. Hall, On a theorem of Frobenius, Proc. London Math. Soc , [5] T. Müller, Modular subgroup arithmetic and a theorem of Philip Hall, Bull. London Math. Soc , [6] T. Müller, Modular subgroup arithmetic, in: Proc Durham Symposium on Groups, Geometries, and Combinatorics A. Ivanov, M. Liebeck, and J. Saxl eds., World Scientific, to appear. SCHOOL OF MATHEMATICAL SCIENCES QUEEN MARY, UNIVERSITY OF LONDON MILE END ROAD LONDON E1 4NS UNITED KINGDOM P.J.Cameron@qmul.ac.uk and T.W.Muller@qmul.ac.uk