有限群在某个极小子群共轭类上的传递性

Transitivity of a finite group on the conjugates of some minimal subgroups.

有限群在某个极小子群共轭类上的某种传递性影响或决定群的构造.运用抽象群和置换群的理论得到:(1)如果有限群G共轭作用在它的所有极小子群上传递,G一定是循环p-群或广义四元数群;(2)如果有限群G在它的某个极小子群共轭类上二重传递,G是某些特殊的群的扩张;(3)如果有限群G是一个几乎单群,G在某个极小子群共轭类上二重传递,G的Socle一定是以下子群之一:PSL(2,p),PSU(3,p2),PSL(2,8).

The transitivity of a finite group on the conjugates of some minimal subgroups influences or determines the structure of the group. Applying the theories of abstract groups and permutation groups, the following results are obtained:(1) If a finite group G acts transitively on all the minimal subgroups by conjugation, G is then a cyclic p-group or a generalized quaternion. (2) If a finite group G acts 2-transitively on the conjugates of some minimal subgroups, G is an extension of some special groups. (3) If G is an almost simple group and acts 2-transitively on the conjugates of some minimal subgroups, its Socle is then one of the following groups: PSL(2,p), PSU (3,p^2), or PSL(2,8).