ON CONJUGACY CLASS SIZES AND CHARACTER DEGREES OF FINITE GROUPS

Let p be a fixed prime, G a finite group and P a Sylow p-subgroup of G. The main results of this paper are as follows: (1) If gcd(p- 1, vertical bar G vertical bar) = 1 and p(2) does not divide vertical bar x(G) | for any p'-element x of prime power order, then G is a solvable p-nilpotent group and a Sylow p- subgroup of G/O-p(G) is elementary abelian. (2) Suppose that G is p-solvable. If p(p-1) does not divide vertical bar x(G) | for any element x of prime power order, then the p-length of G is at most one. (3) Suppose that G is p-solvable. If p(p-1) does not divide chi(1) for any chi is an element of Irr(G), then both the p-length and p'-length of G are at most 2.