Finite groups in which distinct nonlinear irreducible characters have distinct codegrees

Let G be a finite group and let p be a prime. In this paper, we classify all finite quasisimple groups in which the degrees of all irreducible p-Brauer characters are prime powers. As an application, for a fixed odd prime p, we classify all finite nonsolvable groups with the above-mentioned property and having no nontrivial normal p-subgroups. Furthermore, for an arbitrary prime p, a complete classification of finite groups in which the degrees of all nonlinear irreducible p-Brauer characters are primes is also obtained.

On Thompson's Theorem

Squares of characters and finite groups

Let G be a finite group and Irr(G) be the set of irreducible (complex) characters of G. Let χ ∈ Irr(G) and write cod(χ) = |G : ker χ|/χ(1) as the codegree of χ, where ker χ is the kernel of χ. Denote by the intersection of the kernels of all the monolithic, monomial irreducible characters of a p-solvable group G with codegrees divisible by a prime p. We prove in this note that has a normal p-complement, and then the theorem in [2] is generalized. Also, we prove that about the theorem in [2] more is true.

Variants of some of the Brauer-Fowler theorems

A finite group whose irreducible complex characters are rational valued is called a rational group. Thus, G is a rational group if and only if N G(〈x〉)/C G(〈x〉) ≌ Aut(〈x〉) for every x ∈ G. For example, all symmetric groups and their Sylow 2-subgroups are rational groups. Structure of rational groups have been studied extensively, but the general classification of rational groups has not been able to be done up to now. In this paper, we show that a full symmetric group of prime degree does not have any rational transitive proper subgroup and that a rational doubly transitive permutation group containing a full cycle is the full symmetric group. We also obtain several results related to the study of rational groups.

Let $G$ be a finite group with center $Z$8 and irreducible complex character $x$ so that $x{(1)^2} = [G:Z]$, If the $2$-Sylow subgroup of $G/Z$ has order 16 or less then $G$ is solvable.

Let G be a finite group, and let $$\textrm{cd}(G)$$ denote the set of degrees of the irreducible complex characters of G. Define then the character degree graph $$\Delta (G)$$ as the (simple undirected) graph whose vertices are the prime divisors of the numbers in $$\textrm{cd}(G)$$ , and two distinct vertices p, q are adjacent if and only if pq divides some number in $$\textrm{cd}(G)$$ . This paper continues the work, started in [8], toward the classification of the finite non-solvable groups whose degree graph possesses a cut-vertex, i.e. a vertex whose removal increases the number of connected components of the graph. While, in [8], groups with no composition factors isomorphic to $$\textrm{PSL}_{2}(t^a)$$ (for any prime power $$t^a\ge 4$$ ) were treated, here we consider the complementary situation in the case when $$t$$ is odd and $$t^a> 5$$ . The proof of this classification will be then completed in the third and last paper of this series [7] that deals with the case $$t=2$$ .

Simple classical groups of Lie type are determined by their character degrees

For an irreducible character χ of a finite group G, the codegree of χ is defined as . In this paper, we characterize the nonnilpotent groups in which all nonlinear irreducible characters have the same codegree.

Let G be a finite group. The degree (vertex) graph Γ(G) attached to G is a character degree graph. Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adjacent if the greatest common divisor (m, n) > 1. In this paper, we classify all graphs with four vertices that occur as Γ(G) for nonsolvable groups G.

In this paper we study finite p-solvable groups having irreducible complex characters chi in Irr(G) which take roots of unity values on the p-singular elements of G.

Two families of groups close to groups of odd order, and two families of groups close to real groups will be described. The first two are the family of finite groups in which all real irreducible ordinary characters are linear, and the family of all finite groups with the dual condition on the conjugacy classes, namely, groups in which all real conjugacy classes are contained in the center of the group. We will see that each group in one of these families is a direct product of a group of odd order with a 2-group. The families close to real groups are the family of finite groups in which every non-real irreducible ordinary character is linear, and dually, the family of all finite groups in which every non-real conjugacy class is contained in the center. For a group G in the first family we show that the collection of real elements R(G) is a normal subgroupG has a normal π-complement where π = π (∣G: R(G)∣), and a Hall π-subgroup which is either abelian or a nearly real 2-group. A group is in the second family is either abelian, or real or a 2-group. Descriptions and examples of 2-groups satisfying the above conditions will be discussed next. Some of these 2-groups are related to groups treated by several authors. We consider these related groups as well. The interest in the groups mentioned above was triggered by the observation that if G is a finite group in which the square of each irreducible character has at most two irreducible constituents, then all real irreducible characters of G are linear. In the last section of this article we discuss such groups and groups related to them.

Finite groups in which all the nonlinear irreducible characters have equal degrees were described by Isaacs, Passman, and others.The purpose of this article is to consider the other extreme, namely, to characterize all finite groups in which all the nonlinear irreducible characters have distinct degrees.

Perfect nonlinear functions between two finite abelian groups were studied by Carlet and Ding (J Complex 20:205---244, 2004) and Pott (Discret Math Appl 138:177---193, 2004), which can be regarded as a generalization of bent functions on finite abelian groups studied by Logachev et al. (Discret Math Appl 7:547---564, 1997). Poinsot (J Discret Math Sci Cryptogr 9:349---364, 2006), (Cryptogr Commun 4:1---23, 2012) extended this research to arbitrary finite groups, and characterized bent functions on finite nonabelian groups as well as perfect nonlinear functions between two arbitrary finite groups by the Fourier transforms of the related functions at irreducible unitary representations. The purpose of this paper is to study the characterizations of the bentness (perfect nonlinearity) of functions on arbitrary finite groups by the Fourier transforms of the related functions at irreducible characters. We will also give a characterization of a perfect nonlinear function by the relative pseudo-difference family.