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Let G be a finite group, p a prime dividing | G | . In this paper, we study the structure of G, which satisfies the condition that any two irreducible characters in its principal p-block have distinct degrees. We provide a reduction of this question to simple groups and determine the structure of G for p = 2. Furthermore, we describe the structure of G in which any two characters of the principal p-block of G have coprime degrees. [ABSTRACT FROM AUTHOR]

We study and classify a class of representations (called generalized geometric representations) of a Coxeter group of finite rank. These representations can be viewed as a natural generalization of the geometric representation. The classification is achieved by using characters of the integral homology group of certain graphs closely related to the Coxeter graph. On this basis, we also provide an explicit description of those representations on which the defining generators of the Coxeter group act by reflections. [ABSTRACT FROM AUTHOR]

We describe finite groups whose principal block contains only characters of prime power degree. [ABSTRACT FROM AUTHOR]

A group element is called a generalized torsion element if a finite product of its conjugates is equal to the identity. We prove that in a nilpotent or FC-group, the generalized torsion elements are all torsion elements. Moreover, we compute the generalized order of an element in a finite group G using its character table. [ABSTRACT FROM AUTHOR]

In this paper we complete the proof of Brauer's height zero conjecture for two primes proposed by G. Malle and G. Navarro. [ABSTRACT FROM AUTHOR]

The main results of the paper develop a level theory and establish strong character bounds for finite classical groups, in the case that the centralizer of the element has small order compared to | G | in a logarithmic sense. [ABSTRACT FROM AUTHOR]

In this paper, all supercharacter theories of dicyclic groups T 4 n are described. These supercharacter theories are constructed by using the supercharacter theories of dihedral groups of order 4n which are classified by Lamar. [ABSTRACT FROM AUTHOR]

Let G be a finite group and N be a normal subgroup of G. Denote by Irr (G | N) the set of all irreducible complex characters of G whose kernels do not contain N. For χ ∈ Irr (G) , the number cod (χ) = | G : ker (χ) | / χ (1) is called the co-degree of χ . In this paper, we prove that if, for each pair χ , ϕ ∈ Irr (G | N) with cod (χ) ≠ cod (ϕ) , the greatest common divisor of cod (χ) and cod (ϕ) is square-free, then N is solvable. [ABSTRACT FROM AUTHOR]

A finite group G with center Z is of central type if there exists a fully ramified character λ ∈ Irr (Z) , i. e. the induced character λ G is a multiple of an irreducible character. Howlett–Isaacs have shown that G is solvable in this situation. A corresponding theorem for p-Brauer characters was proved by Navarro–Späth–Tiep under the assumption that p ≠ 5 . We show that there are no exceptions for p = 5 , i. e. every group of p-central type is solvable. Gagola proved that every solvable group can be embedded in G/Z for some group G of central type. We generalize this to groups of p-central type. As an application we construct some interesting non-nilpotent blocks with a unique Brauer character. This is related to a question by Kessar and Linckelmann. [ABSTRACT FROM AUTHOR]

The Suzuki simple group Sz(8) has an automorphism group 3. Using the electronic Atlas [22], the group Sz(8) : 3 has an absolutely irreducible module of dimension 12 over F 2. Therefore a split extension group of the form 2 12 : (S z (8) : 3) : = G ¯ exists. In this paper we study this group, where we determine its conjugacy classes and character table using the coset analysis technique together with Clifford-Fischer Theory. We determined the inertia factor groups of G ¯ by analysing the maximal subgroups of Sz(8) : 3 and maximal of the maximal subgroups of Sz(8) : 3 together with various other information. It turns out that the character table of G ¯ is a 43 × 43 complex valued matrix, while the Fischer matrices are all integer valued matrices with sizes ranging from 1 to 7. [ABSTRACT FROM AUTHOR]

For a finite group G, we denote by μ (G) and c(G), the minimal degree of faithful permutation representation of G, and the minimal degree of faithful representation of G by quasi-permutation matrices over the complex field C , respectively. In this article, we study μ (G) and c(G) for various classes of finite non-abelian p-groups with cyclic center. We prove a result for normally monomial p-groups with cyclic center which generalizes a result of Behravesh for finite p-groups of nilpotency class 2 with cyclic center (J. London Math. Soc.55(02) (1997) 251–260, Theorem 4.12). We also compute minimal degrees for some classes of metabelian p-groups. [ABSTRACT FROM AUTHOR]

For a positive integer l and a finite group G, let S n m l (G) = Σ χ ∈ Irr n m (G) χ l (1) Σ χ ∈ Irr n m (G) χ l − 1 (1) , where Irr n m (G) denotes the set of non-monomial characters of G. In this paper, we prove that if S n m l (G) < S n m l ( SL 2 (5)) , then G is solvable. In particular, we show that for a non-solvable group G, if S n m l (G) = S n m l ( SL 2 (5)) , then G = MC is a central product of two subgroups M and C such that M ≅ SL 2 (5) , Z (M) = M ∩ C and G/M is an M-group. Among the other results, we arrive at the solvability of G by assuming Σ χ ∈ Irr n m (G) χ (1) > 9 | G | / 29 . [ABSTRACT FROM AUTHOR]

Using methods from multilinear algebra, orbital subspaces of n-fold tensor spaces have been realized as group modules in the literature. Such subspaces, however, are nearly isometric to group modules already known as coset spaces. By determining the group module structure of coset spaces, we offer an alternative derivation of the group module structure of orbital subspaces. [ABSTRACT FROM AUTHOR]

We improve a few results related to Huppert's ϱ-σ conjecture. We also generalize a result about the covering number of character degrees to arbitrary finite groups. [ABSTRACT FROM AUTHOR]

Let G be a finite group and p be a prime number dividing the order of G. An irreducible character χ of G is called a quasi p-Steinberg character if χ(g) is nonzero for every p-regular element g in G. In this paper, we classify the quasi p-Steinberg characters of complex reflection groups G(r,q,n) and exceptional complex reflection groups. In particular, we obtain this classification for Weyl groups of type Bn and type Dn. [ABSTRACT FROM AUTHOR]

Let G be a finite group and χ be an irreducible character of G, the number cod (χ) = | G : ker (χ) | / χ (1) is called the codegree of χ. Also, cod(G) = {cod(χ) | χ ∈Irr(G)}. For d ∈cod(G), the multiplicity of d in G, denoted by m G ′ (d) , is the number of irreducible characters of G having codegree d. A finite group G is called a T k ′ -group for some integer k ≥ 1, if there exists d0 ∈cod(G) such that m G ′ (d 0) = k and for every d ∈cod(G) −{d0}, we have m G ′ (d) = 1 . In this note we characterize finite T k ′ -groups completely, where k ≥ 1 is an integer. [ABSTRACT FROM AUTHOR]

We determine all possible values of the integer group determinant of C 2 4 , where C 2 is the cyclic group of order 2. [ABSTRACT FROM AUTHOR]

Let C be the pseudovariety F of all finite groups or the pseudovariety S of all finite solvable groups and let Γ be either a finitely generated free group or a surface group. The C -genus of Γ , denoted by G C (Γ) , consists of the isomorphism classes of finitely generated residually- C groups G having the same quotients in C as Γ . We show that the groups from G C (Γ) are residually-p for all primes p. This answers a question of Gilbert Baumslag and shows that the groups in the genus are residually finite rationally solvable groups. This leads to a positive solution of particular case of a question of Alexander Grothendieck: if F is a free group, G is a finitely generated residually- C group and u : F → G is a homomorphism such that the induced map of pro- C completions u C ^ : F C ^ → G C ^ is an isomorphism, then u is an isomorphism. [ABSTRACT FROM AUTHOR]

Let GG be a π\pi -separable group for some set π\pi of primes, let χ∈Bπ(G)\chi \in {B}_{\pi }\left(G) and let N◃GN\hspace{0.33em}\triangleleft \hspace{0.33em}G. In this article, we explore how to construct a nucleus for an irreducible constituent of χN{\chi }_{N} via the given nucleus (W,γ)\left(W,\gamma ) for χ\chi .

Let $\Gamma $ be a finite group, let $\theta $ be an involution of $\Gamma $ and let $\rho $ be an irreducible complex representation of $\Gamma $ . We bound ${\operatorname {dim}} \rho ^{\Gamma ^{\theta }}$ in terms of the smallest dimension of a faithful $\mathbb {F}_p$ -representation of $\Gamma /\operatorname {\mathrm {Rad}}_p(\Gamma )$ , where p is any odd prime and $\operatorname {\mathrm {Rad}}_p(\Gamma )$ is the maximal normal p-subgroup of $\Gamma $ .

In this paper, we strengthen a result of Moretó and Wolf on the divisibility of the degree of a product of irreducible characters by the index of the Fitting subgroup in a solvable group. We also strengthen some results about Gluck's conjecture to monomial characters. [ABSTRACT FROM AUTHOR]

We give a full list of the orthogonal determinants of the even degree indicator '+' ordinary irreducible characters of SL 3 (q) and SU 3 (q) . [ABSTRACT FROM AUTHOR]

Let G be a finite group. A collection Π = { H 1 , ⋯ , H r } of subgroups of G, where r > 1 , is said a non-trivial partition of G if every non-identity element of G belongs to one and only one H i , for some 1 ⩽ i ⩽ r . We call a group G that does not admit any non-trivial partition a partition-free group. In this paper, we study a partition-free group G whose all proper non-cyclic subgroups admit non-trivial partitions. [ABSTRACT FROM AUTHOR]

Gaschütz (1954) proved that a finite group G has a faithful irreducible complex representation if and only if its socle is generated by a single element as a normal subgroup; this result extends to arbitrary fields of characteristic p so long as G has no nontrivial normal p-subgroup. žmud′ (1956) showed that the minimum number of irreducible constituents in a faithful complex representation of G coincides with the minimum number of generators of its socle as a normal subgroup; this result can also be extended to arbitrary fields of any characteristic p such that G has no nontrivial normal p-subgroup (i.e., over which G admits a faithful completely reducible representation). Rhodes (1969) characterized the finite semigroups admitting a faithful irreducible representation over an arbitrary field as generalized group mapping semigroups over a group admitting a faithful irreducible representation over the field in question. Here, we provide a common generalization of the theorems of žmud′ and Rhodes by determining the minimum number of irreducible constituents in a faithful completely reducible representation of a finite semigroup over an arbitrary field (provided that it has one). Our key tool for the semigroup result is a relativized version of žmud′'s theorem that determines, given a finite group G and a normal subgroup N ◃ G , what is the minimum number of irreducible constituents in a completely reducible representation of G whose restriction to N is faithful. [ABSTRACT FROM AUTHOR]

Given a finite group G, we study the monoid algebra RG, over a field K , generated by the monomial characters of G. In particular, we note that the integral closure of RG is contained in the algebra generated by those characters χ for which their associated Artin L-function L(s, χ) is holomorphic at s0 ∈ ℂ \ {1}. Also, we discuss the supercharacter theoretic case. [ABSTRACT FROM AUTHOR]

Let G be a finite group. Denoting by cd (G) the set of the degrees of the irreducible complex characters of G, we consider the character degree graph of G: this, is the (simple, undirected) graph whose vertices are the prime divisors of the numbers in cd (G) , and two distinct vertices p, q are adjacent if and only if pq divides some number in cd (G) . This paper completes the classification, started in Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. II, 2022. https://doi.org/10.1007/s10231-022-01299-3) and Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119), of the finite non-solvable groups whose character degree graph has a cut-vertex, i.e., a vertex whose removal increases the number of connected components of the graph. More specifically, it was proved in Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119 that these groups have a unique non-solvable composition factor S, and that S is isomorphic to a group belonging to a restricted list of non-abelian simple groups. In Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. II, 2022. https://doi.org/10.1007/s10231-022-01299-3) and Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119) all isomorphism types for S were treated, except the case S ≅ PSL 2 (2 a) for some integer a ≥ 2 ; the remaining case is addressed in the present paper. [ABSTRACT FROM AUTHOR]

We study a class of finite groups, called almost monomial groups, which generalize the class of monomial groups and is connected with the theory of Artin L-functions. Our method of research is based on finding similarities with the theory of monomial groups, whenever it is possible. [ABSTRACT FROM AUTHOR]

For a character χ of a finite group G the number cod(χ) = |G: Ker(χ)|/χ(1) is called the codegree of χ. In this note, we verify the equivalent version of Huppert's conjecture for K3 -groups. [ABSTRACT FROM AUTHOR]