群${\mathbb{Z}}_n^k$的自同构个数的均值

On the Average Number of Automorphisms of the Group ${\mathbb{Z}}_n^k$

摘要: 令${\mathbb{Z}}_n$ 表示模$n$ 剩余类群，${\mathbb{Z}}_n^k$ 表示$k$ 个${\mathbb{Z}}_n$ 做直积后的群，并记其自同构个数为$a_k\left(n\right)$ 。对任意的正整数$n$ 以及任意给定的正整数$k$ ，我们得到了${\displaystyle {\sum }_{n\le x}{a}_k\left(n\right)}$ 的渐近公式，其表明$a_k\left(n\right)$ 的平均个数在不计常数因子的意义下是$n^{k^2}$ 。该结果可被视作是Euler函数的经典均值结果在群论意义下的推广。

Abstract: Let ${\mathbb{Z}}_n$ be the additive group of the residue classes modulo $n$ , ${\mathbb{Z}}_n^k$ be the direct product of $k$ ${\mathbb{Z}}_n$ ’s, and $a_k\left(n\right)$ be the number of automorphisms of ${\mathbb{Z}}_n^k$ . For arbitrary positive integer $n$ and any given positive integer $k$ , we obtain the asymptotic formula for the sum ${\displaystyle {\sum }_{n\le x}{a}_k\left(n\right)}$ , which indicates that the average value of $a_k\left(n\right)$ is $n^{k^2}$ up to multiplying by a constant. This is a generalization, from a group theory viewpoint, of a classic result for the mean value of the Euler totient function.