有限交换群的直积分解

Direct Product of Finite Abelian Groups

已知有限交换群的最高阶元生成的循环群是其直积因子.主要得到了:有限交换群的最低阶生成元生成的子群也是其直积因子.即设G是交换群,x是群G的最低阶生成元,则存在子群G1≤G,使得G=〈x〉×G1.

In a finite abelian group ,the cyclic subgroup generated by a maximal order element is its direct product .Here ,we show that the subgroup generated by a generated minimal order element is also a direct product ,that is ,let G be an abelian group and x a generated element of minimal order in G ,then there exists G1 ≤G such that G=× G1 .