关于中心外的同阶元共轭的有限群的注记

Remarks on the finite groups with conjugate non-central same order elements

令G是一个非Abel的有限群,并设G的中心外的同阶元是共轭的.钱国华等证明了GS3,但证明很复杂且依赖有限单群分类定理.如其所述,定理的证明是否可以避免对于有限单群分类定理的依赖是一个值得关注的问题.本文在不依赖有限单群分类定理的条件下讨论对G的Sylow 2-子群加以某个限制的几种情形的证明,如当G的Sylow 2-子群是Abel群时的证明.

Let Gbe a finite non-abelian group.It is proved by Qian et al.that if non-central same order elements in Gare conjugate,then G≌S3.In order to prove this theorem,the theorem for the classification of finite simple groups is used.In this paper,we discuss the proof of some special cases that Sylow2-subgroups of Gsatisfy some condition without this theorem.For example,we prove that if Gis a finite non-abelian group with abele Sylow 2-subgroups and non-central same order elements in Gare conjugate,then G≌S3.