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 $G = N \rtimes Q$ 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.