关于Zm子群的一个注记

On Subgroups of Z*m

讨论 n整除 m时 Z*m 的子群与 Z*n 的子群之间的关系以及它们对有限群的 Galois作用的不同 ,并给出相关的例子 .证明当 α≥ 3 ,β≥ 1时 ,Z*2 α 的一个子群与 Z*2 α+β 的一个子群在 mod2α下同构当且仅当该子群为〈-1〉;而对于奇素数 p及α,β≥ 1 ,Z*pα 的一个子群与 Z*pα+β 的一个子群在 mod pα下同构当且仅当该子群的阶是 p-1的因子 .

This note considers some relationships between subgroups of Z * m and those of Z * n providing m is divided by n exactly. We get that,for α≥3,β≥1,a subgroup of Z * 2 α is isomorphic to some subgroup of Z * 2 α+β under mod 2 α if and only if which is <-1>,and that,for an odd prime p and α,β≥1,a subgroup of Z * p α is isomorphic to some subgroup of Z * p α+β under mod p α if and only if its order is a factor of p-1.