The automorphism group of an abelian $p$-group and its normal $p$-subgroups

Abstract:

Let ${\mathbf {\Gamma }}$ be the automorphism group of a nonelementary reduced abelian p-group, $p \geq 5$. It is shown that every noncentral normal subgroup of ${\mathbf {\Gamma }}$ contains a noncentral normal subgroup ${\mathbf {\Delta }}$ of ${\mathbf {\Gamma }}$ such that ${{\mathbf {\Delta }}^p} = 1$. Furthermore, every cyclic normal subgroup of ${\mathbf {\Gamma }}$ is contained in the center of ${\mathbf {\Gamma }}$.