The largest Irreducible Character Degree of a Finite Group

Abstract

Much information about a finite group is encoded in its character table. Indeed even a small portion of the character table may reveal significant information about the group. By a famous theorem of Jordan, knowing the degree of one faithful irreducible character of a finite group gives an upper bound for the index of its largest normal abelian subgroup.Here we consider b(G), the largest irreducible character degree of the group G. A simple application of Frobenius reciprocity shows that b(G) ≧ |G:A| for any abelian subgroup A of G. In light of this fact and Jordan's theorem, one might seek to bound the index of the largest abelian subgroup of G from above by a function of b(G). If is G is nilpotent, a result of Isaacs and Passman (see [7, Theorem 12.26]) shows that G has an abelian subgroup of index at most b(G)4.

Large character degrees of solvable groups with abelian Sylow 2-subgroups

“…In [7] D. Gluck proves that in all finite groups the index of the Fitting subgroup F(G) in G is bounded by a polynomial function of b(G). For solvable G, Gluck further shows that [G : F(G)] b(G) 13/2 and conjectures that [G : F(G)] b(G) 2 ; this has been verified by A. Espuelas [4] for G of odd order.…”

“…Following Gluck's strategy in [7] for producing irreducible characters of large degree, we consider the action of G/ F(G) on the group V of the linear characters of the section F(G)/Φ(G). V is a faithful and completely reducible G/ F(G)-module and by standard Clifford theory large orbits of G/ F(G) on V give correspondingly large character degrees.…”

Two Applications of Lewis' Theorem on Character Degree Graphs of Solvable Groups

“…We write b(G) for the largest irreducible character degree of G. Gluck conjectured [3] that the inequality |G :…”

Irreducible characters of even degree and normal Sylow 2-subgroups

“…Together with Proposition 2.3, this result allows us to bound the number of (strongly real) irreducible characters of small degrees of a finite group with a nonabelian minimal normal subgroup, and then to control the invariant acd 2 (G) of such a group, see Section 2 and Proposition 3.1. We hope that the techniques developed here will be useful in the future study of other problems involving the average degree of a certain set of characters and other invariants concerning character degrees like the largest character degree [3,7,8,10] or the character degree ratio [2,16].…”