Some Notes On The Proof Of Hall-Higman-Shult Type Theorem

Aiming to simplify the work of the Classification of Finite Simple groups,Flavell has extended a fixed point theorem of Shult to arbitrary finite groups without the framework of it,and he also gives a criterion for a weakly closed subgroup to be strongly closed.Based on his paper,we give two results on finite group theory including a group isomorphic to 2-dimensional special linear group provided some hypothesis,a lower bound controlled by a function of the order of its center for the order of a 2-group,and we introduce a very useful criterion for controlling a subgroup to be strong closed and strongly fusions of a subgroup on some conditions.Also,we construct an exceptional case mentioned in theorem A from Flavell's paper and give some details for his proofs.