FIXED CHARACTERS FOR /^-GROUPS OF OPERATORS R. GOW Suppose H is a group of automorphisms of the finite group G. It is possible to define actions of //on both the conjugacy classes and the complex characters of G, and when / / i s cyclic, a lemma of Brauer, [4; p. 536], shows that there is equality between the number of fixed points of these actions. When H is not cyclic, further conditions are needed on H to ensure the validity of the above equality, as an example in [5] indicates. Glauberman has shown in [2; Corollary 9] that if //is solvable and G and H are of coprime order, then the desired equality holds. In [5], Isaacs obtained informa- tion on the fixed classes and characters when G and H are not of coprime order, under the additional assumptions that G is solvable and H has a normal /^-complement for each prime/? dividing the order of G. In this paper we suppose that H is a /?-group, which is a particular case of the situation dealt with by Isaacs. However, assuming only that G is /^-solvable, we are able to obtain more precise information on the number of fixed points in the two permutation actions of H. We deduce our results from a lemma concerning the number of irreducible /^-modular characters of G fixed by H. The lemma requires no solvability conditions on G and is essentially a repeated application of the original lemma of Brauer. The hypothesis that G is/7-solvable is required to enable us to apply the Fong-Swan theorem and thus to gain information on the complex characters fixed by H. We recall here two pertinent definitions concerning modular representations. Firstly, a /7-regular class of a group is a conjugacy class whose members are of order coprime to p. Secondly, an absolutely irreducible /^-modular character, 6, of a finite group is the character of an irreducible representation of the group defined over an algebraically closed field of characteristic/?. Thus 6 takes its values in some finite field of characteristic p and is distinct from the complex-valued Brauer character of the representation. With these definitions we may proceed to the proof of the lemma from which our main theorem is deduced. LEMMA. Let G be a finite group which admits the p-group H as a group of auto- morphisms. The number of p-regular classes of G fixed by H equals the number of absolutely irreducible p-modular characters of G fixed by H. Proof We proceed by induction on the order of H. When H has order 1, our theorem is simply a restatement of a basic result of Brauer [1; p. 591]. Thus we can suppose that H has order at least/;. Let M be a subgroup of index p in H. We form the semi-direct product of G with M, the action of M on G being that induced by the action of H on G. We can assume that the lemma is true for the action of M on G. Thus if M fixes exactly s /^-regular classes of G, it fixes exactly s irreducible modular characters <f> i} ..., <[> s of G. Now as M is & p-group and the <j>i are defined over afieldof characteristic/?, it is a consequence of a result of Green, [3; Lemma 2.2.3], and the Received 5 June, 1974; revised 7 November, 1974. [J. LONDON MATH SOC. (2), 12 (1976), 281-283]