On the Rarity of Quasinormal Subgroups

Abstract

For each prime $p$ and positive integer $n$, Berger and Gross have defined a finite $p$-group $G=HX$, where $H$ is a core-free quasinormal subgroup of exponent $p^{n-1}$ and $X$ is a cyclic subgroup of order $p^n$. These groups are universal in the sense that any other finite $p$-group, with a similar factorisation into subgroups with the same properties, embeds in $G$. In our search for quasinormal subgroups of finite $p$-groups, we have discovered that these groups $G$ have remarkably few of them. Indeed when $p$ is odd, those lying in $H$ can have exponent only $p$, $p^{n-2}$ or $p^{n-1}$. Those of exponent $p$ are nested and they all lie in each of those of exponent $p^{n-2}$ and $p^{n-1}$.