Study On The Length Multiplicities Of Conjugacy Classes Of Finite Simple Groups

This thesis mainly investigates finite simple groups whose conjugacy class lengths satisfy some conditions.First,we prove that any finite simple group has two different conjugacy classes of same size.As a consequence,if each of conjugacy classes of a finite nonabelian group has different size,then the group is not simple.In addition,we also prove the following result:Let G be a finite nonabelian simple group.If the 2'-part of the greatest common divisor of any two of different conjugacy class lengths of G is square-free,then G is J1,2B2(q)with(q-1)(q2+1)being square-free,or G-A1(q)and one of the following holds:(1)q is an odd prime and((q-1)(q+1)2'is square-free.(2)q=2f?4 and((q-1)(q+1?2'is square-free.