A Note on the Action of ^-groups on Abelian Groups

Let £-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 p1'. If A is a p'group, then a simple application of Corollary 2.4 of [2] yields \p\ ^p2e and | JP| ikp21 ■ On the other hand, if A is not a p'-group, then it is not true that \P\ is bounded by a function of pe or pf alone. However, \P\ is bounded by a function of both pe and pf and we show in fact that \p\ ^oh-dV+d2. 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 xtyj = yj if ij^j and Xiyi = yiyoIt is easy to see that all orbit sizes in A are at most p. On the other hand, if \EA does not contain y0 in its kernel then X has pn conjugates. Thus e = l and/ = ra and \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 / are reversed. Thus/=1, e = n and \P\ is not bounded by a function of pf. Example 2. Let A be elementary abelian of order pn+1 viewed as a vector space of dimension ra + 1 over GF(£). Let P be a Sylow psubgroup of Aut(^4)^GL(ra + l, p). Then \p\ = £«(-+«/» and P can be represented as the set of all lower triangular matrices over GFip) with all diagonal entries equal to 1. This follows easily by order considerations. Let xEP and aEA. Then xa=a if and only if (x —l)a = 0. Let a be fixed. Then its centralizer in P is the solution space of ra homogeneous equations in ra(ra + l)/2 unknowns. Thus | (Sp(fl) | g£l(i/«n<n+i)-n) anc} [p. gp(a) ] g£». Hence all orbits have size at most pn and it is easy to see that at least one orbit has size p". Thus e = n. Since A=A and P is a Sylow ^-subgroup of GL(w + l, p), it follows that also/=ra. Therefore the exponent of p in \p\ is essentially a quadratic function of e and / Since the general bound obtained in this paper is essentially biquadratic, there would appear to be room for improvement.