## 2012.96: On $p$-soluble groups with a generalized $p$-central or powerful Sylow $p$-subgroup

2012.96:
E. I. Khukhro
(2012)
*On $p$-soluble groups with a generalized $p$-central or powerful Sylow $p$-subgroup.*

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## Abstract

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.

Item Type: | MIMS Preprint |
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Uncontrolled Keywords: | p-length; powerful Sylow p-subgroup; generalized p-central p-group |

Subjects: | MSC 2000 > 20 Group theory and generalizations |

MIMS number: | 2012.96 |

Deposited By: | Professor Evgeny Khukhro |

Deposited On: | 18 October 2012 |

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