| 王玉雷,刘合国.广义超特殊p-群的自同构群Ⅲ[J].数学年刊A辑,2011,32(3):307~318 |
| 广义超特殊p-群的自同构群Ⅲ |
| The Automorphism Group of a Generalized Extraspecial p-group III |
| |
| DOI： |
| 中文关键词: 广义超特殊p-群 中心积 辛群 自同构 |
| 英文关键词:Generalized extraspecial p-group Central product Symplectic group Automorphism |
| 基金项目:国家自然科学基金(No.10971054); 河南省教育厅自然科学基金(No.2011B110011); 河南工业大学科研基金(No.10XZZ011);河南工业大学引进人才专项基金(No.2009BS029)资助的项目 |
|
| Hits: 1752 |
| Download times: 912 |
| 中文摘要: |
| 确定了广义超特殊$p$-群$G$的自同构群的结构. 设$|G|=p^{2n+m}$, $|\zeta G|=p^m$, 其中$n \geq 1$, $m \geq 2$,
${\rm Aut}_{f}G$是${\rm Aut}\, G$中平凡地作用在${\rm Frat}\, G$上的元素形成的正规子群, 则
(1) 当$G$的幂指数是$p^m$时,
(i) 如果$p$是奇素数, 那么${\rm Aut}\, G/{\rm Aut}_{f}G\cong\mathbb{Z}_{(p-1)p^{m-2}}$, 并且
${\rm Aut}_{f}G/{\rm Inn}\, G\cong{\rm Sp}(2n,p)\times\mathbb{Z}_p$.
(ii) 如果$p=2$, 那么${\rm Aut}\, G={\rm Aut}_{f}G$\ (若$m=2$)或者
${\rm Aut}\, G/{\rm Aut}_{f}G\cong\mathbb{Z}_{2^{m-3}}\times\mathbb{Z}_2$\ (若$m \geq 3$),
并且${\rm Aut}_{f}G/{\rm Inn}\, G\cong{\rm Sp}(2n,2)\times\mathbb{Z}_2$.
(2) 当$G$的幂指数是$p^{m+1}$时,
(i) 如果$p$是奇素数, 那么${\rm Aut}\, G=\langle\theta\rangle\ltimes{\rm Aut}_{f}G$, 其中$\theta$的阶是$(p-1)p^{m-1}$,
且${\rm Aut}_{f}G/{\rm Inn}\, G\cong K\rtimes{\rm Sp}(2n-2,p)$, 其中$K$是$p^{2n-1}$阶超特殊$p$-群.
(ii) 如果$p=2$, 那么${\rm Aut}\, G=\langle\theta_1, \theta_2\rangle\ltimes{\rm Aut}_{f}G$, 其中$\langle\theta_1, \theta_2\rangle=
\langle\theta_1\rangle\times\langle\theta_2\rangle\cong\mathbb{Z}_{2^{m-2}}\times\mathbb{Z}_2$,
并且${\rm Aut}_{f}G/{\rm Inn}\, G\cong K\rtimes{\rm Sp}(2n-2,2)$, 其中$K$是$2^{2n-1}$阶初等Abel $2$-群.
特别地, 当$n=1$时, ${\rm Aut}_{f}G/{\rm Inn}\, G\cong\mathbb{Z}_{p}$. |
| 英文摘要: |
| In this paper, the automorphism group of a generalized extraspecial $p$-group $G$ is determined, where $p$ is a prime number. Assume that
$|G|=p^{2n+m}$ and $|\zeta G|=p^m$, where $n \geq 1$ and $m \geq 2$. Let ${\rm Aut}_{f}G$ be the normal subgroup of ${\rm Aut}\, G$
consisting of all elements of ${\rm Aut}\, G$ which act trivially on ${\rm Frat}\, G$. Then
(1) When the exponent of $G$ is equal to $p^m$,
(i) If $p$ is odd, then ${\rm Aut}\, G/{\rm Aut}_{f}G\cong\mathbb{Z}_{(p-1)p^{m-2}}$ and ${\rm Aut}_{f}G/{\rm Inn}\, G\cong{\rm Sp}(2n,p)\times\mathbb{Z}_p$.
(ii) If $p=2$, then ${\rm Aut}\, G={\rm Aut}_{f}G$
(when $m=2$) or ${\rm Aut}\, G/{\rm Aut}_{f}G\cong\mathbb{Z}_{2^{m-3}}\times\mathbb{Z}_2$ (when $m\geq 3$),
and ${\rm Aut}_{f}G/{\rm Inn}\, G\cong{\rm Sp}(2n,2)\times\mathbb{Z}_2$.
(2) When the exponent of $G$ is equal to $p^{m+1}$,
(i) If $p$ is odd, then ${\rm Aut}\, G=\langle\theta\rangle\ltimes{\rm Aut}_{f}G$, where $\theta$ is of order $(p-1)p^{m-1}$, and
${\rm Aut}_{f}G/{\rm Inn}\, G\cong K\rtimes{\rm Sp}(2n-2,p)$, where $K$ is an extraspecial $p$-group of order $p^{2n-1}$.
(ii) If $p=2$, then ${\rm Aut}\, G=\langle\theta_1, \theta_2\rangle\ltimes{\rm Aut}_{f}G$, where
$\langle\theta_1,
\theta_2\rangle=\langle\theta_1\rangle \times\langle\theta_2\rangle\cong\mathbb{Z}_{2^{m-2}}\times\mathbb{Z}_2$,
and ${\rm Aut}_{f}G/{\rm Inn}\, G\cong K\rtimes{\rm Sp}(2n-2,2)$, where $K$ is an elementary abelian 2-group of order $2^{2n-1}$.
In particular, ${\rm Aut}_{f}G/{\rm Inn}\, G\cong\mathbb{Z}_{p}$ when $n=1$. |
| View Full Text View/Add Comment Download reader |
| Close |
|
|
|
|
|
