On the normal subgroups of a split extension

Abstract

Let $N$ and $Q$ be a finite groups with $\gcd (|N|,|Q|)=1$. In this paper we describe normal subgroups of $G=N⋊Q$ via normal subgroups of $N$ and $Q$. Let $p$ and $q$ be distinct primes. Let ${\mathfrak{A}}_p$ be the variety of elementary abelian $p$-groups. Let ${\mathfrak{A}}_p{\mathfrak{A}}_q$ be the variety of extensions of a group in ${\mathfrak{A}}_p$ by a group in ${\mathfrak{A}}_q$. We also provide a method of determining the normal subgroups of a group in the variety ${\mathfrak{A}}_p{\mathfrak{A}}_q$. We also provide the complete list of normal subgroups of a finite group with cyclic Sylow subgroups.

10 pages