Finite groups satisfying character degree congruences

Let G be a finite group, p a prime and {c\in\{0,1,\ldots,p-1\}} . Suppose that the degree of every nonlinear irreducible character of G is congruent to c modulo p. If here {c=0} , then G has a normal p-complement by a well known theorem of Thompson. We prove that in the cases where {c\neq 0} the group G is solvable with a normal abelian Sylow p-subgroup. If {p\neq 3} then this is true provided these character degrees are congruent to c or to {-c} modulo p.

Groups whose bipartite divisor graph for character degrees has four or fewer vertices

Let [Formula: see text] be a finite group and [Formula: see text] be the set of nonlinear irreducible character degrees of [Formula: see text]. Suppose that [Formula: see text] is the set of primes dividing some elements of [Formula: see text]. The bipartite divisor graph for [Formula: see text], [Formula: see text], is a graph whose vertices are the disjoint union of [Formula: see text] and [Formula: see text], and a vertex [Formula: see text] is connected to a vertex [Formula: see text] if and only if [Formula: see text]. In this paper, we consider groups whose graph has four or fewer vertices. We show that all these groups are solvable and determine the structure of these groups. We also provide examples of any possible graph.

A nilpotency criterion related to the minimal degree of a nonlinear irreducible character

Let [Formula: see text] be a finite nonabelian group, let [Formula: see text] be the minimal degree of a nonlinear irreducible character of [Formula: see text] and suppose that [Formula: see text] for some positive integer [Formula: see text]. Then [Formula: see text] is nilpotent.

Coprime Group Actions Fixing All Nonlinear Irreducible Characters

The main result of this paper is the following:Theorem A. Let H and N be finite groups with coprime orders andsuppose that H acts nontrivially on N via automorphisms. Assume that Hfixes every nonlinear irreducible character of N. Then the derived subgroup ofN is nilpotent and so N is solvable of nilpotent length≦ 2.Why might one be interested in a situation like this? There has been considerable interest in the question of what one can deduce about a group Gfrom a knowledge of the setcd(G) = ﹛x(l)lx ∈ Irr(G) ﹜of irreducible character degrees of G.Recently, attention has been focused on the prime divisors of the elements of cd(G). For instance, in [9], O. Manz and R. Staszewski consider π-separable groups (for some set π of primes) with the property that every element of cd(G) is either a 77-number or a π'-number.