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...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
University of Isfahan
2012-06-01
|
| Series: | International Journal of Group Theory |
| Subjects: | |
| Online Access: | http://www.theoryofgroups.ir/?_action=showPDF&article=761&_ob=a54f9c582725efbbb45bb105f241bdb7&fileName=full_text.pdf |
| _version_ | 1818289985248296960 |
|---|---|
| author | Evgeny Khukhro |
| author_facet | Evgeny Khukhro |
| author_sort | Evgeny Khukhro |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-13T02:20:58Z |
| format | Article |
| id | doaj.art-e6792affde364c8ea9b89d0ed00bf38e |
| institution | Directory Open Access Journal |
| issn | 2251-7650 2251-7669 |
| language | English |
| last_indexed | 2024-12-13T02:20:58Z |
| publishDate | 2012-06-01 |
| publisher | University of Isfahan |
| record_format | Article |
| series | International Journal of Group Theory |
| spelling | doaj.art-e6792affde364c8ea9b89d0ed00bf38e2022-12-22T00:02:46ZengUniversity of IsfahanInternational Journal of Group Theory2251-76502251-76692012-06-01125157On p-soluble groups with a generalized p-central or powerful Sylow p-subgroupEvgeny KhukhroLet $G$ 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.http://www.theoryofgroups.ir/?_action=showPDF&article=761&_ob=a54f9c582725efbbb45bb105f241bdb7&fileName=full_text.pdfp-central p-group of height kpowerful p-groupp-solublep-length |
| spellingShingle | Evgeny Khukhro On p-soluble groups with a generalized p-central or powerful Sylow p-subgroup International Journal of Group Theory p-central p-group of height k powerful p-group p-soluble p-length |
| title | On p-soluble groups with a generalized p-central or powerful Sylow p-subgroup |
| title_full | On p-soluble groups with a generalized p-central or powerful Sylow p-subgroup |
| title_fullStr | On p-soluble groups with a generalized p-central or powerful Sylow p-subgroup |
| title_full_unstemmed | On p-soluble groups with a generalized p-central or powerful Sylow p-subgroup |
| title_short | On p-soluble groups with a generalized p-central or powerful Sylow p-subgroup |
| title_sort | on p soluble groups with a generalized p central or powerful sylow p subgroup |
| topic | p-central p-group of height k powerful p-group p-soluble p-length |
| url | http://www.theoryofgroups.ir/?_action=showPDF&article=761&_ob=a54f9c582725efbbb45bb105f241bdb7&fileName=full_text.pdf |
| work_keys_str_mv | AT evgenykhukhro onpsolublegroupswithageneralizedpcentralorpowerfulsylowpsubgroup |