| 王玉雷,刘合国.关于一类中心循环的有限p-群的自同构群[J].数学年刊A辑,2017,38(2):191~200 |
| 关于一类中心循环的有限p-群的自同构群 |
| On the Automorphism Group of a Class of Finite p-Groups with Cyclic Center |
| Received:April 11, 2015 Revised:August 20, 2016 |
| DOI：10.16205/j.cnki.cama.2017.0015 |
| 中文关键词: Finite $p$-groups, Cyclic centers, Automorphism groups |
| 英文关键词:Finite $p$-groups, Cyclic centers, Automorphism groups |
| 基金项目:本文受到国家自然科学基金(No.11301150,No.11371124)
和河南省自然科学基金(No.142300410134,No.162300410066)的资助. |
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| 中文摘要: |
| 确定了一类中心循环的有限$p${-}群$G$的自同构群. 设$G=X_{3}(p^m)^{*n}*\mathbb{Z}_{p^{m+r}}$,
其中$m\geq 1,\ n\geq 1$和$r\geq 0$, 并且
$$
X_3(p^m)=\langle x,y \mid x^{p^m}=y^{p^m}=1,\ [x,y]^{p^m}=1,\ [x,[x,y]]=[y,[x,y]]=1\rangle.
$$
$\mathrm{Aut}_n G$表示$\mathrm{Aut}\,G$中平凡地作用在$N$上的元素形成的正规子群,
其中$G'\leq N \leq \zeta G,\ |N|=p^{m+s},\ 0\leq s\leq r$, 则
(i) 如果$p$是一个奇素数, 那么$\mathrm{Aut}\,G/\mathrm{Aut}_nG\cong \mathbb{Z}_{p^{m+s-1}(p-1)}$,
$\mathrm{Aut}_nG/\mathrm{Inn}\,G\cong \mathrm{Sp}(2n,\mathbb{Z}_{p^m})$\linebreak $\times
\mathbb{Z}_{p^{r-s}}$.
(ii) 如果$p=2$, 那么$\mathrm{Aut}\,G/\mathrm{Aut}_nG\cong H$,
其中$H=1$ (当$m+s=1$时)或者$\mathbb{Z}_{2^{m+s-2}}\times\mathbb{Z}_2$ (当$m+s\geq 2$时).
进一步地,
$\mathrm{Aut}_nG/\mathrm{Inn}\,G\cong K\times L$,
其中$K=\mathrm{Sp}(2n,\mathbb{Z}_{2^m})$ (当$r>0$时)或者$\mathrm{O}(2n,\mathbb{Z}_{2^m})$ (当$r=0$时),
$L=\mathbb{Z}_{2^{r-1}}\times \mathbb{Z}_2$ (当$m=1,\ s=0,\ r\geq 1$时)或者$\mathbb{Z}_{2^{r-s}}$. |
| 英文摘要: |
| The automorphism group of a class of finite $p$-groups with cyclic center
is determined. Let $G=X_{3}(p^m)^{*n}*\mathbb{Z}_{p^{m+r}}$,
where $m\geq 1,\ n\geq 1,\ r\geq 0$, and
$$
X_3(p^m)=\langle x,y \mid x^{p^m}=y^{p^m}=1,\ [x,y]^{p^m}=1,\ [x,[x,y]]=[y,[x,y]]=1\rangle.
$$
Let $\mathrm{Aut}_n G$ be the normal
subgroup of $\mathrm{Aut}\,G$ consisting of all elements of
$\mathrm{Aut}\,G$ which act trivially on $N$, where $G'\leq N \leq
\zeta G$ and $|N|=p^{m+s},\ 0\leq s \leq r$. Then
(i) If $p$ is odd, then $\mathrm{Aut}\,G/\mathrm{Aut}_nG\cong \mathbb{Z}_{p^{m+s-1}(p-1)}$ and $\mathrm{Aut}_nG/\mathrm{Inn}\,G\cong \mathrm{Sp}(2n,\mathbb{Z}_{p^m})\times
\mathbb{Z}_{p^{r-s}}$.
(ii) If $p=2$, then $\mathrm{Aut}\,G/\mathrm{Aut}_nG\cong H$,
where $H=1$ (if $m+s=1$) or $\mathbb{Z}_{2^{m+s-2}}\times\mathbb{Z}_2$ (if $m+s\geq 2$). Furthermore,
$\mathrm{Aut}_nG/\mathrm{Inn}\,G\cong K\times L$,
where $K=\mathrm{Sp}(2n,\mathbb{Z}_{2^m})$ (if $r>0$) or $\mathrm{O}(2n,\mathbb{Z}_{2^m})$ (if $r=0$),
$L=\mathbb{Z}_{2^{r-1}}\times \mathbb{Z}_2$ (if $m=1,\ s=0,\ r\geq 1$) or $\mathbb{Z}_{2^{r-s}}$. |
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