| 徐涛,刘合国,余杨.关于有限Abelp-群的自同构群[J].数学年刊A辑,2019,40(2):199~210 |
| 关于有限Abelp-群的自同构群 |
| On the Automorphism Groups of Finite Abelian p-Groups |
| Received:March 26, 2016 Revised:December 27, 2016 |
| DOI：10.16205/j.cnki.cama.2019.0016 |
| 中文关键词: Finite abelian $p$-group, Automorphism group, Frattini subgroup |
| 英文关键词:Finite abelian $p$-group, Automorphism group, Frattini subgroup |
| 基金项目:本文受到国家自然科学基金(No.11626078,No.11371124)和河北省教育厅青年基金(No.QN2016184)的资助. |
| Author Name | Affiliation | E-mail | | XU Tao | Department of Science, Hebei University of Engineering,Handan 056038, Hebei, China. | gtxutao@163.com | | LIU Heguo | College of Mathematics and Statistics, Hubei University,Wuhan 430062, China. | ghliu@hubu.edu.cn | | YU Yang | College of Mathematics and Statistics, Hubei University,Wuhan 430062, China. | 459193638@qq.com |
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| 中文摘要: |
| 从有限Abel~$p${-}群$P$的型不变量出发, 给出了其自同构群$\mbox{Aut}P$的阶的计算公式,
并利用$|\mbox{Aut}P|$的计算公式得到了下面3个结果:
1. 由有限Abel~$p${-}群的型不变量的两种变换得到了其自同构群的阶的变化规律;
2. 用群的阶、 秩、 幂指数三个量界定了有限Abel~$p${-}群的自同构的阶;
3. 对部分$\mbox{Frattini}$子群为$p$阶群的有限$p${-}群, 确定了其自同构群的阶何时达到最小值和最大值. |
| 英文摘要: |
| Starting from the invariant of a finite abelian $p$-group $P$,
the authors obtain the computational formula of the order of its automorphism group $\mbox{Aut}P$.
Three applications of this computational formula are given as follows.
Firstly, they find some properties on the order of its automorphism group from
two transformations of invariant of a finite abelian $p$-group.
Secondly, they estimate the order of automorphism of a finite abelian $p$-group by a
function depending on order, rank and exponent
of this group. Thirdly, letting $P$ be a finite $p$-group with Frattini subgroup of prime
order, they give the conditions to guarantee the order of $\mbox{Aut}P$ attains the maximal value or minimal value, respectively. |
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