Research On Finite Simple Groups Any Two Of Whose Different Conjugacy Class Lengths Have Almost Prime Power Greatest Common Divisor

This thesis is mainly to study finite simple groups in which the greatest common divisor of any pair of different conjugacy class lengths is a prime power or the double of a prime power,and in which the total number of(not necessarily different)prime divisors of the greatest common divisor of any pair of different conjugacy class lengths is at most 2.It is proved that the only finite simple groups satisfying the above conditions are A1(4),A1(8)and A1(q),where q is an odd prime power satisfying(1)q-1=2f,q+1=2·rm,where r is a prime,f,m?N+,or(2)q+1=2f,q-1=2·m,where r is a prime and f,m?N+;and A1(4)(?)A1(5)(?)A5,respectively.