Largest Character Degree Research Articles

On large orbits of subgroups of linear groups

The main aim of this paper is to prove an orbit theorem and to apply it to obtain a result that can be regarded as a significant step towards the solution of Gluck’s conjecture on large character degrees of finite solvable groups.

Recognition of $${\mathrm {PSL}}(2, p^{2})$$ PSL ( 2 , p 2 ) by Order and Some Properties on Character Degrees

Let G be a finite group and $${\mathrm {Irr}}(G)$$ be the set of all irreducible complex characters of G. Furthermore $${\mathrm {cd}}(G)$$ is the set of all character degrees of G. In this paper, we introduce a new characterization of $${\mathrm {PSL}}(2, p^{2})$$ , where p is an odd prime number, by order and some properties on character degrees. In fact, we prove that if $$|G |=| {\mathrm {PSL}}(2, p^{2})| $$ and $$p^{2}+1 \in {\mathrm {cd}}(G)$$ and there exists no $$\theta \in {\mathrm {Irr}}(G)$$ such that $$2p \mid \theta (1)$$ , then $$G \cong {\mathrm {PSL}}(2, p^{2}) $$ . Also by an example we show that $$ {\mathrm {PSL}}(2, p^{2}) $$ , where p is an odd prime, is not recognizable by order and the largest character degree.

The largest character degree, conjugacy class size and subgroups of finite groups

ABSTRACTLet G be a finite group, let b(G) denote the largest irreducible character degree of the group G and let bcl(G) denote the largest conjugacy class size of the group G. We study the relations between the sizes of the nilpotent and solvable subgroups of G and b(G). We also study the relations between the sizes of the nilpotent and solvable subgroups of G and bcl(G).

The largest character degree and the Sylow subgroups of finite groups

Let [Formula: see text] be a Sylow [Formula: see text]-subgroup and [Formula: see text] be the largest irreducible character degree of a finite nonabelian group [Formula: see text]. Then [Formula: see text].

Bounding group orders by large character degrees: A question of Snyder

Abstract Let G be a nonabelian finite group and let d be an irreducible character degree of G. Then there is a positive integer e so that |G| = d(d + e). Snyder has shown that if e > 1, then |G| is bounded by a function of e. This bound has been improved by Isaacs and by Durfee and Jensen. In this paper we will show, for groups having a nontrivial, abelian normal subgroup, that |G| ≤ e 4 - e 3. Given that there are a number of solvable groups that meet this bound, it is best possible. Our work makes use of results regarding Camina pairs, Gagola characters, and Suzuki 2-groups.

BOUNDING THE NUMBER OF IRREDUCIBLE CHARACTER DEGREES OF A FINITE GROUP IN TERMS OF THE LARGEST DEGREE

We conjecture that the number of irreducible character degrees of a finite group is bounded in terms of the number of prime factors (counting multiplicities) of the largest character degree. We prove that this conjecture holds when the largest character degree is prime and when the character degree graph is disconnected.

Large character degrees of solvable 3’-groups

We prove that if G G is a finite solvable group and 3 ∤ | G : F ( G ) | 3 \nmid |G:\mathbf {F}(G)| , then the index of the Fitting subgroup of G G is at most the square of the largest irreducible character degree of G G .

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

We prove that if G is a solvable group with abelian Sylow 2-subgroups, the index of the Fitting subgroup of G is at most the square of the largest irreducible character degree of G.

The largest character degree and the Sylow subgroups of finite groups*1

Let P be a Sylow p -subgroup and b ( G ) the largest irreducible character degree of a finite group G . It is shown that |P/O p (G)| <b(G) in general and | P / O p ( G )|⩽ b ( G ) in case P is abelian.

The largest character degree and the Sylow subgroups of finite groups

Let P be a Sylow p-subgroup and b( G) the largest irreducible character degree of a finite group G. It is shown that |P/O p(G)| <b(G) in general and | P/ O p ( G)|⩽ b( G) in case P is abelian.

Large character degrees of groups of odd order

Large character degrees of groups of odd order