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 pe, (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 pe> p but none is divisible by pe+1. In
each of these situations, group theoretic information is deduced from the
character theoretic hypothesis and in several cases complete characterizations are
obtained.
