Characterization of Finite Simple Groups With Abelian Sylow 2-Subgroups

Abstract

For the non-Abelian simple groups with Abelian Sylow 2-subgroups J. H. Walterhas proved the following famous theorem. Lemma 1. If F is a non-Ablian simple group with Abelian Sylow 2-subgroups, thenone of the following holds:(i)F≌PSL(2,Q),q>3,q≡3,5(mod 8) or q=2~n,n≥2;(ii)F≌J_1;(iii)F≌R(q),q=3~(2m+1),m≥1.Let G be a finite group and let π_e(G) denote the set of all orders of elements in