关于循环群和交换群的等价刻画

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.

Equivalent characterizations of cyclic groups and abelian groups

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.