Decomposition of a Group over an Abelian Normal Subgroup

Abstract

Let a group G have an Abelian normal subgroup A; put \( \overline{G} \) = G/A and \( \overline{g} \) = gA for g ∈ G. We can think of A as a right ℤ\( \overline{G} \)-module and define the action of an element \( u={\alpha}_1{\overline{g}}_1 \) +…+ \( {\alpha}_n{\overline{g}}_n \) ∈ ℤ\( \overline{G} \) on a ∈ A by a formula a^u = \( {\left({a}^{g_1}\right)}^{\alpha_1} \) ·…· \( {\left({a}^{g_n}\right)}^{\alpha_n} \) ; here \( {a}^{g_i}={g}_i^{-1}a{g}_i \). Denote by \( {\Theta}_{\mathbb{Z}\overline{G}} \)(A) the annihilator of A in the ring ℤ\( \overline{G} \), which is a two-sided ideal. Let \( R=\mathbb{Z}\overline{G}/{\Theta}_{\mathbb{Z}\overline{G}}(A) \). 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 \( \overline{G} \)·D, where D is an R-module. It is also proved that an R-decomposition always exists in one important case.