Decomposition of a Group over an Abelian Normal Subgroup

Let a group G have an Abelian normal subgroup A ; put G ¯ = G/A and g ¯ = gA for g ∈ G . We can think of A as a right ℤ G ¯ -module and define the action of an element u = α 1 g ¯ 1 +…+ α n g ¯ n ∈ ℤ G ¯ on a ∈ A by a formula a u = a g 1 α 1 ·…· a g n α n ; here a g i = g i − 1 a g i . Denote by Θ ℤ G ¯ ( A ) the annihilator of A in the ring ℤ G ¯ , which is a two-sided ideal. Let R = ℤ G ¯ / Θ ℤ G ¯ A . A subgroup A can also be treated as an R -module. We give a criterion for the existence of an R -decomposition of G over A , i.e., the possibility of embedding G in a semidirect product G ¯ · D , where D is an R -module. It is also proved that an R -decomposition always exists in one important case.