IMPLICATIONS OF PERMUTIZERS OF SOME SUBGROUPS IN FINITE GROUPS.

The purpose of this paper is to investigate the influence of permutizers of some subgroups of finite groups on the structure of finite groups. We use permutizers of maximal subgroups as well other simple conditions to characterize supersolubility of finite groups. Also we use permutizers of almost maximal subgroups to get a necessary and sufficient condition for a finite group to be supersoluble, which shows that the permutizer condition Is, in a sense, equivalent to the condition that P_G(M)>M if M is an almost maximal subgroup of G. The paper generalizes and deepens the related results in Beidleman and Robinson (1997), Zhang (1986), and Guo (1989). [ABSTRACT FROM AUTHOR]