In several previous papers I.
M. Isaacs and this author studied properties of groups which are related to the
degrees of their absolutely irreducible representations and in particular to the biggest
such degree. The results were concerned mainly with the existence of “large”
abelian subgroups in these groups. It was found that much more could be
said in the p-group-like situation in which the degrees of the irreducible
characters of group G are all powers of a fixed prime p. We say group G has r.x.e
(representation exponent e) if the degrees of all the irreducible characters of
G divide pe. In this paper we characterize groups with r.x.2. It is found
that the prime p = 2 plays a special role here. This supports the conjecture
that additional and more complicated groups with r.x.e occur for p ≦ e.
With a few exceptions for p = 2, all groups G with r.x.2 are shown to have
either a normal subgroup of index p with r.x.1 or a center of index dividing
p6.
