On $p$-soluble groups with a generalized $p$-central or powerful Sylow $p$-subgroup

Khukhro, E. I. (2012) On $p$-soluble groups with a generalized $p$-central or powerful Sylow $p$-subgroup. [MIMS Preprint]

Let $G$ be a finite $p$-soluble group, and $P$ a Sylow $p$-sub\-group of $G$. It is proved that if all elements of $P$ of order $p$ (or of order ${}\leq 4$ for $p=2$) are contained in the $k$-th term of the upper central series of $P$, then the $p$-length of $G$ is at most $2m+1$, where $m$ is the greatest integer such that $p^m-p^{m-1}\leq k$, and the exponent of the image of $P$ in $G/O_{p',p}(G)$ is at most $p^m$. It is also proved that if $P$ is a powerful $p$-group, then the $p$-length of $G$ is equal to 1.