Vol. 15, No. 3, 1965

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

I. Martin (Irving) Isaacs and Donald Steven Passman

Vol. 15 (1965), No. 3, 877–903

The purpose of this paper is to study properties of groups which are related to the degrees of their absolutely irreducible characters and in particular to the biggest such degree.

Let p be a fixed prime. We say group G has r.x.e. (representation exponent e) if the degrees of all the absolutely irreducible characters of G divide pe. If G has a subinvariant abelian subgroup whose index divides pe then by Ito’s Theorem, G has r.x.e. We show conversely that if G has r.x.e. then G has a subinvariant abelian subgroup whose index divides p4e. While we do not obtain the best possible value for the exponent in the above bound, we do show that it is essentially a linear function of e.

We can obtain information about somewhat larger subgroups. We show that a group G with r.x.e. has a subgroup H of index pe with [H : Z(H)] p3e(e+2), where Z(H) is the center of H. The latter bound is by no means best possible. However we show by example that a similar result cannot hold in general for subgroups of index less than pe.

We study the case e = 1 in more detail and completely characterize all such groups. This generalizes a result of Amitsur which discusses the p = 2 situation. We prove that G has r.x.1 if and only if (i) G is abelian, (ii) G has a normal abelian subgroup of index p or (iii) [G : Z(G)] = p3.

The previous results apply to rather special groups. We consider the more general case now. We say group G has r.b.n. (representation bound n) if the degrees of all the absolutely irreducible characters of G are n. If G has an abelian subgroup A with [G : A] n then as is easily seen G has r.b.n. Conversely we show here that there is a finite valued function h with the property that if G has r.b.n. then G has an abelian subgroup A with [G : A] h(n). This result can be viewed as an analog to Jordan’s Theorem for complex linear groups of degree n.

The analogy between groups with r.b.n. and linear groups of degree n can be carried further. We show that if G has r.b.n. and if p is a prime with p > n then G has a normal abelian Sylow p-subgroup.

Finally we discuss some extensions of the above results to infinite discrete groups.

Mathematical Subject Classification
Primary: 20.80
Received: 17 March 1964
Published: 1 September 1965
I. Martin (Irving) Isaacs
Donald Steven Passman