A characterization of groups in terms of the degrees of their characters. II

Abstract

In this paper we continue our study of the relationship between the structure of a finite group G and the set of degrees of its irreducible complex characters. The following hypotheses on the degrees are considered: (A) G has r.x. e for some prime p, i.e. all the degrees divide p e , (B) the degrees are linearly ordered by divisibility and all except 1 are divisible by exactly the same set of primes, (C) G has a.c. m, i.e., all the degrees except 1 are equal to some fixed m, (D) all the degrees except 1 are prime (not necessarily the same prime) and (E) all the degrees except 1 are divisible by p e > p but none is divisible by p e+1 . In each of these situations, group theoretic information is deduced from the character theoretic hypothesis and in several cases complete characterizations are obtained.