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 $p$-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 gre...
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| Format: | Article |
| Language: | English |
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University of Isfahan
2012-06-01
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| Series: | International Journal of Group Theory |
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| Online Access: | http://www.theoryofgroups.ir/?_action=showPDF&article=761&_ob=a54f9c582725efbbb45bb105f241bdb7&fileName=full_text.pdf |
| Summary: | 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. |
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| ISSN: | 2251-7650 2251-7669 |