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

Document Type: Research Paper

Author

Sobolev Institute of Mathematics, Novosibirsk

Abstract

Let $G$ be a finite $p$-soluble group‎, ‎and $P$ a Sylow $p$-subgroup 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.

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