交换群和循环群的若干充分必要条件

利用交换子群的中心化子和正规化子对有限群结构的强的控制作用,通过限制二元生成交换子群、初等交换子群、极大交换子群、循环子群、极小子群等的中心化子一致于正规化子,得到交换群和循环群的7个充分必要条件,改进了Zassenhaus定理和陈重穆在文献2]中提出的定理0.3.

Some Necessary and Sufficient Conditions of Abelian Groups and Cyclic Groups

The centralizers and normalizers of abelian subgroups play a strong part in the structure of finite groups.In this paper,we consider some abelian subgroups,such as abelian subgroups generated by two elements,elementary abelian subgroups,maximal abelian subgroups,cyclical subgroups,minmal subgroups,whose centralizers are equal to its normalizers,so we obtain some necessary and sufficient conditions of abelian groups and cyclic groups,and improve Zassenhaus Theorem and Chen Zhongmu,Theorem 0.3 in Reference 2].