Khukhro, E. (2012). On $p$-soluble groups with a generalized $p$-central or powerful Sylow $p$-subgroup. International Journal of Group Theory, 1(2), 51-57. doi: 10.22108/ijgt.2012.761

Evgeny Khukhro. "On $p$-soluble groups with a generalized $p$-central or powerful Sylow $p$-subgroup". International Journal of Group Theory, 1, 2, 2012, 51-57. doi: 10.22108/ijgt.2012.761

Khukhro, E. (2012). 'On $p$-soluble groups with a generalized $p$-central or powerful Sylow $p$-subgroup', International Journal of Group Theory, 1(2), pp. 51-57. doi: 10.22108/ijgt.2012.761

Khukhro, E. On $p$-soluble groups with a generalized $p$-central or powerful Sylow $p$-subgroup. International Journal of Group Theory, 2012; 1(2): 51-57. doi: 10.22108/ijgt.2012.761

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

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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