G-Groups and Biuniform Abelian Normal Subgroups

We prove a weak form of the Krull-Schmidt Theorem concerning the behavior of direct-product decompositions of G-groups, biuniform abelian G-groups, G-semi-direct products and the G-set Hom(H,A). Here G and A are groups and H is a G-group. Our main result is the following. Let P be any group. Let H-1, ... ,H-n, H'(1), ... ,H'(t) be n+t biuniform abelian normal subgroups of P. Suppose that the products H-1, ... ,H-n, H'(1), ... ,H'(t) are direct, that is, H-n, ... ,H(1 )x ... x H-n and H'(1), ... ,H't = H'(t) x ... x H'(t) .Then the normal subgroups H(1 )x ... x H-n and H'(1) x ... x H'(t) of P are P-isomorphic if and only if n = t and there exist two permutations sigma - and tau of {1, 2, ... , n} such that [H-i](m) = [H'(sigma(i))](m) and [H-i](e) = [H'(tau(i))](e) for every i = 1, 2, ... ,n.