同阶交换子群个数之集为{1,3}的有限群

Finite Groups Whose Set of the Number of Abelian Subgroups of the Same Order Is{1,3}

证明了不存在同阶交换子群个数之集为{1,2}的有限群,并且完全确定了同阶交换子群个数之集为{1,3}的有限群结构.作为推论,得到:群G的同阶交换子群个数之集为{1,3}等价于群G的同阶子群个数之集为{1,3}.

It is proved in this paper that there is no finite group G satisfying the condition that the set of the number of abelian subgroups of the same order is{1,2}.Furthermore,it is determined that the structure of the finite group G whose set of the number of abelian subgroups of the same order is{1,3}.It is,hence,derived that for a group G,the set of the number of abelian subgroups of the possible order is{1,3}if and only if the set of the number of subgroups of the possible order is{1,3}.