Finite p-central groups of height k

Abstract

A finite group $G$ is called {\it $p^i$-central of height $k$} if every element of order $p^i$ of $G$ is contained in the $k^{th}$-term $\zeta_k(G)$ of the ascending central series of $G$. If $p$ is odd such a group has to be $p$-nilpotent (Thm. A). Finite $p$-central $p$-groups of height $p-2$ can be seen as the dual analogue of finite potent $p$-groups, i.e., for such a finite $p$-group $P$ the group $P/\Omega_1(P)$ is also $p$-central of height $p-2$ (Thm. B). In such a group $P$ the index of $P^p$ is less or equal than the order of the subgroup $\Omega_1(P)$ (Thm. C). If the Sylow $p$-subgroup $P$ of a finite group $G$ is $p$-central of height $p-1$, $p$ odd, and $N_G(P)$ is $p$-nilpotent, then $G$ is also $p$-nilpotent (Thm. D). Moreover, if $G$ is a $p$-soluble finite group, $p$ odd, and $P\in \text{Syl}_p(G)$ is $p$-central of height $p-2$, then $N_G(P)$ controls $p$-fusion in $G$ (Thm. E). It is well-known that the last two properties hold for Swan groups.