Groups of p-rank 2 containing an isolated element of order p

Abstract

Let p be an odd prime and G be a finite group with O_{p'}(G)=1 of p-rank at most 2 that contains an isolated element of order p. If xnotin Z(G), we show that F^*(G) is simple and we describe the structure of a Sylow p-subgroup P of F^*(G) as well as the fusion system mathcal F_P(F^*(G)) without using the classification of finite simple groups.