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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.
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