关于循环群和交换群的等价刻画

Equivalent characterizations of cyclic groups and abelian groups

考虑某些交换子群具有特殊的正规化子,用初等方法证明了循环群和交换群的等价刻画:设G为有限群,则G是循环群当且仅当G的每个极小子群的正规化子皆是循环群;G是交换群当且仅当G的每个初等交换子群的正规化子皆是交换群.

Considering some abelian subgroups with special normalizers,we have proved the following equivalent characterizations of cyclic groups and abelian groups by elementary methods:Let G be a finite group;then G is a cyclic group if and only if the normalizer of every minimal subgroup of G is a cyclic group;G is an abelian group if and only if the normalizer of every elementary abelian subgroup of G is an abelian group.