p-Groups of automorphisms of abelianp-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 Ω₁(A) = {x ε A|px=0}; the kernels are denoted by C_G(A/pA) and C_G(Ω₁(A)), respectively. It is proved that if the exponent of one of the kernels C_G(A/pA) or C_G(Ω₁(A)) is finite then the other also has a finite exponent bounded in terms of the first; moreover, these kernels are nilpotent. In one case we impose the additional restriction . And the wreath product 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].) Translated fromAlgebra i Logika, Vol. 39, No. 3, pp. 359–371, May–June, 2000.