Decomposition of a group over an Abelian normal subgroup

N. S. Romanovskii

Algebra i logika, Tome 55 (2016) no. 4, pp. 478-492

Résumé

Let a group $G$ have an Abelian normal subgroup $A$; put $\overline G=G/A$ and $\overline g=gA$ for $g\in G$. We can think of $A$ as a right $\mathbb Z\overline G$-module and define the action of an element $u=\alpha_1\overline g_1+\dots+\alpha_n\overline g_n\in\mathbb Z\overline G$ on $a\in A$ by a formula $a^u=(a^{g_1})^{\alpha_1}\cdot\ldots\cdot(a^{g_n})^{\alpha_n}$; here $a^{g_i}=g^{-1}_iag_i$. Denote by $\Theta_{\mathbb Z\overline G}(A)$ the annihilator of $A$ in the ring $\mathbb Z\overline G$, 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 $\overline G\cdot D$, where $D$ is an $R$-module. It is also proved that an $R$-decomposition always exists in one important case.

Détail

Keywords: Abelian normal subgroup
Mots-clés : $R$-decomposition.