A note on the action of -groups on abelian groups

Let p-group P act faithfully on abelian group A. Then P also acts faithfully on A, the abelian group of all linear complex characters of A. Suppose that all orbits in A under the action of P have size at most pe and that all orbits in A have size at most pf. If A is a pgroup, then a simple application of Corollary 2.4 of [2] yields I PI ?p2e and IPI p2f. On the other hand, if A is not a p'-group, then it is not true that IPI is bounded by a function of pe or pf alone. However, IPI is bounded by a function of both pe and pf and we show in fact that PI P2(e+1)2(f+1)2 We first discuss several examples. EXAMPLE 1. Let A be elementary abelian of order pn+l generated by yo, yi, * * *, y. and let P be elementary abelian of order pn generated by xi, * * , xn. Define the action of P on A by xiyj=yj if i j and xiyi = yiyo. It is easy to see that all orbit sizes in A are at most p. On the other hand, if XGA` does not contain yo in its kernel then X has pn conjugates. Thus e = 1 and f = n and I P is not bounded by a function of pe. If we consider the induced action of P on A, then the roles of e and f are reversed. Thus f = 1, e = n and I P I is not bounded by a function of pf. EXAMPLE 2. Let A be elementary abelian of order pn+l viewed as a vector space of dimension n+1 over GF(p). Let P be a Sylow psubgroup of Aut(A)_GL(n+1, p). Then IPI =pn(n+l)/2 and P can be represented as the set of all lower triangular matrices over GF(p) with all diagonal entries equal to 1. This follows easily by order considerations. Let x CP and a CA. Then xa = a if and only if (x -1)a = O. Let a be fixed. Then its centralizer in P is the solution space of n homogeneous equations in n(n+l)/2 unknowns. Thus | Sp(a)| >p(l1/2)n(n+1)-n} and [P: Sp(a) ] ? pn. Hence all orbits have size at most pn and it is easy to see that at least one orbit has size pn. Thus e=n. Since A=A and P is a Sylow p-subgroup of GL(n+l, p), it follows that also f= n. Therefore the exponent of p in IPI is essentially a quadratic function of e and f. Since the general bound obtained in this paper is essentially biquadratic, there would appear to be room for improvement.