On the Automorphism Groups of Finite Abelian p-Groups

DOI：10.16205/j.cnki.cama.2019.0016

 作者 单位 E-mail 徐涛 河北工程大学数理学院, 河北 邯郸 056038. gtxutao@163.com 刘合国 湖北大学数学与统计学院, 武汉 430062. ghliu@hubu.edu.cn 余杨 湖北大学数学与统计学院, 武汉 430062. 459193638@qq.com

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.

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.