Vol. 17, No. 3, 1966

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Groups whose irreducible representations have degrees dividing p2

Donald Steven Passman

Vol. 17 (1966), No. 3, 475–496

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.

Mathematical Subject Classification
Primary: 20.80
Received: 26 December 1964
Published: 1 June 1966
Donald Steven Passman