关于有限Abel p-群的自同构群

On the Automorphism Groups of Finite Abelian p-Groups

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.

Starting from the invariant of a finite abelian p-group P, the authors obtain the computational formula of the order of its automorphism group AutP. 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 AutP attains the maximal value or minimal value, respectively.