关于有限Abel p-群的自同构群

从有限Abel p-群P的型不变量出发,给出了其自同构群AutP的阶的计算公式,并利用|AutP|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.

On the Automorphism Groups of Finite Abelian p-Groups

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.