p-GROUPS OF AUTOMORPHISMS OF ABELIAN p-GROUPS

We consider the action of a p-group G on an Abelian p-group A. with the latter treated as a faithful right ZG-module. Our aim is to establish a connection between exponents of the kernels under the induced action of G on elementary p-groups A/pA and (A) = {x(epsilon) A Ipx = 0}; the kernels are denoted by CG(A/pA) and C(G)(A)), respectively. It is proved that if the exponent of one of the kernels C(G)(A/pA) orC(G)(A)) is finite then the other also has afinite exponent bounded in terms of the first; moreover, these kernels are nilpotent. In one case we impose the additional restriction p^i A = 0. And the wreath product Cp I G of a quasicyclic group and an arbitrary p-group G shows that this condition cannot be dropped. The results obtained are used to confirm, for one particular case, the conjecture on the boundedness of a derived length of a finite group with an automorphism of order 2 all of whose fixed points are central. (The solubility of such groups, and also the reduction to the case of 2-groups, were established in [1].)