The subclass algebra associated with a finite group and subgroup

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.

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.

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.

In the general case, the order of a finite nonidentity group G is substantially larger than the squared degree of every irreducible character Θ of G, i.e., Θ(1)2< G. In the present paper, we study finite groups with an irreducible character Θ such that $$\left| G \right| \leqslant 2\Theta {\left( 1 \right)^2}$$ .

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

Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

Let G be a finite group and let w A IrrðGÞ lie in a p-block of full defect for an odd prime p. We investigate mQðwÞp, the p-part of the rational Schur index of w. At the onset of this work, it was suspected that these Schur indices would not be divisible by p, and in Section 2 below, we show this to be the case when G is either supersolvable or p-nilpotent. Further motivation came from the situation in which G has a normal Sylow p-subgroup. Of course in this case all irreducible characters of G lie in p-blocks of full defect. Recall that a p-hyperelementary subgroup is a subgroup with a cyclic, normal p-complement and that by results of Brauer, Witt, and Berman G has a p-hyperelementary subgroup H with z A IrrðHÞ such that mQðwÞp divides mQðzÞ. If the Sylow p-subgroup of G is normal, all p-hyperelementary subgroups of G are nilpotent. Hence, the Schur indices of their irreducible characters are at most 2, and it follows that p does not divide mQðwÞ: One final case is that in which the Sylow p-subgroups of G are abelian. In this situation, Brauer’s height zero conjecture implies that the degrees of the irreducible characters in p-blocks of full defect are not divisible by p. Thus their Schur indices are not divisible by p either. However, in Section 3 we fix an a A N and construct a solvable group G with irreducible character w lying in the principal p-block of G=KerðwÞ and satisfying mQðwÞ 1⁄4 p. It is crucial to regard w as a faithful irreducible character of G=KerðwÞ, as one can always regard an irreducible character c of a group H as an irreducible character of a larger group G which surjects onto H. This does not a¤ect the Schur index of c, but it can easily be set up to ensure c lies in the principal p-block of G: The group SLð2; 3Þ has a rational-valued irreducible character w such that

Applications of Frobenius Algebras to Representation Theory of Schur Algebras

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.

It is a well-known fact that characters of a finite group can give important information of the structure of the group. Also it was proved by the second author that a finite simple group can uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite group by using less information of its character table, and successfully characterize K 3-groups by their orders and one or two irreducible character degrees of their character tables.

Gagola and Lewis proved that a finite group [Formula: see text] is nilpotent if and only if [Formula: see text] divides [Formula: see text] [Formula: see text] [Formula: see text] for all irreducible characters [Formula: see text] of [Formula: see text]. In this paper, we prove that a finite soluble group [Formula: see text] is nilpotent if and only if [Formula: see text] divides [Formula: see text] [Formula: see text] [Formula: see text] for all irreducible monomial characters [Formula: see text] of [Formula: see text].

For a character χ of a finite group G, the number is called the codegree of χ. Let N be a normal subgroup of G and set Let p be a prime. In this paper, we first show that if for two distinct prime divisors p and q of , divides none of the codegrees of elements of , then and N is either p‐solvable or q‐solvable. Next, we classify the finite groups with exactly one irreducible character of the codegree divisible by p and, also finite groups whose codegrees of irreducible characters which are divisible by p are equal. Then, we prove that p‐length of a finite p‐solvable group is not greater than the number of the distinct codegrees of its irreducible characters which are divisible by p. Finally, we consider the case when the codegree of every element of is square‐free.

Let n, r ∈ ℕ. The affine Schur algebra (of type A) over a field K is defined to be the endomorphism algebra of certain tensor space over the extended affine Weyl group of type A r−1. By the affine Schur–Weyl duality it is isomorphic to the image of the representation map of the action on the tensor space when K is the field of complex numbers. We show that can be defined in another two equivalent ways. Namely, it is the image of the representation map of the semigroup algebra (defined in Section 3) action on the tensor space, and it equals to the “dual” of a certain formal coalgebra related to this semigroup. By these approaches, we can show many relations between different Schur algebras and affine Schur algebras and reprove one side of the affine Schur–Weyl duality.

We say that a finite group $G$ is conjugacy expansive if for any normal subset $S$ and any conjugacy class $C$ of $G$ the normal set $SC$ consists of at least as many conjugacy classes of $G$ as $S$ does. Halasi, Mar'oti, Sidki, Bezerra have shown that a group is conjugacy expansive if and only if it is a direct product of conjugacy expansive simple or abelian groups. By considering a character analogue of the above, we say that a finite group $G$ is character expansive if for any complex character $alpha$ and irreducible character $chi$ of $G$ the character $alpha chi$ has at least as many irreducible constituents, counting without multiplicity, as $alpha$ does. In this paper we take some initial steps in determining character expansive groups.

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 $$t=2$$ .