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Given a finite group G and an irreducible complex character χ of G , the codegree of χ is defined as the integer cod (χ) = | G : ker (χ) | / χ (1). It was conjectured by G. Qian in [16] that, for every element g of G , there exists an irreducible character χ of G such that cod (χ) is a multiple of the order of g ; the conjecture has been verified under the assumption that G is solvable ([16]) or almost-simple ([13]). In this paper, we prove that Qian's conjecture is true for every finite group whose Fitting subgroup is trivial, and we show that the analysis of the full conjecture can be reduced to groups having a solvable socle. [ABSTRACT FROM AUTHOR]

Let G be a finite group, and p a prime number; a character of G is called p-constant if it takes a constant value on all the elements of G whose order is divisible by p. This is a generalization of the very important concept of characters of p-defect zero. In this paper, we characterize the finite p-solvable groups having a faithful irreducible character that is p-constant and not of p-defect zero, and we will show that a non- p-solvable group with this property is an almost-simple group. [ABSTRACT FROM AUTHOR]

Let G be a finite group, and let cs (G) denote the set of sizes of the conjugacy classes of G. The prime graph built on cs (G) , that we denote by Δ (G) , is the (simple undirected) graph whose vertices are the prime divisors of the numbers in cs (G) , and two distinct vertices p , q are adjacent if and only if pq divides some number in cs (G). A rephrasing of the main theorem in [8] is that the complement Δ ‾ (G) of the graph Δ (G) does not contain any cycle of length 3. In this paper we generalize this result, showing that Δ ‾ (G) does not contain any cycle of odd length , i.e., it is a bipartite graph. In other words, the vertex set V (G) of Δ (G) is covered by two subsets, each inducing a complete subgraph (a clique). As an immediate consequence, setting ω (G) to be the maximum size of a clique in Δ (G) , the inequality | V (G) | ≤ 2 ω (G) holds for every finite group G. [ABSTRACT FROM AUTHOR]

Let $G$ be a finite group and $p$ a prime. We say that a $p$-regular element $g$ of $G$ is $p$-nonvanishing if no irreducible $p$-Brauer character of $G$ takes the value $0$ on $g$. The main result of this paper shows that if $G$ is solvable and $g\in G$ is a $p$-regular element which is $p$-nonvanishing, then $g$ lies in a normal subgroup of $G$ whose $p$-length and $p^{\prime }$-length are both at most 2 (with possible exceptions for $p\leq 7$), the bound being best possible. This result is obtained through the analysis of one particular orbit condition in linear actions of solvable groups on finite vector spaces, and it generalizes (for $p>7$) some results in Dolfi and Pacifici [‘Zeros of Brauer characters and linear actions of finite groups’, J. Algebra 340 (2011), 104–113]. [ABSTRACT FROM AUTHOR]

Abstract: We describe the finite groups whose p-Brauer character table, for or , does not contain any zero. This completes the analysis in [6], where we considered the case . [Copyright &y& Elsevier]

Abstract: Let G be a finite group, and p a prime number greater than 3. It is known that, if every irreducible p-Brauer character of G does not vanish on any -element of G, then G is solvable. The primary aim of this work is to describe the structure of groups satisfying the above condition; among other more specific properties, we show that the -length of G is at most 2 (the bound being the best possible). The structural results are obtained as an application of the main theorem in this paper, that deals with particular linear actions of solvable groups on finite vector spaces. [Copyright &y& Elsevier]

Let G be a finite group. An element g ∈ G is called a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g) = 0. In this paper we study the vanishing prime graph Γ(G), whose vertices are the prime numbers dividing the orders of some vanishing element of G, and two distinct vertices p and q are adjacent if and only if G has a vanishing element of order divisible by pq. Among other things we prove that, similarly to what holds for the prime graph of G, the graph Γ(G) has at most six connected components. [ABSTRACT FROM PUBLISHER]

Let G be a finite group, and Irr( G) the set of irreducible complex characters of G. We say that an element g ∈ G is a vanishing element of G if there exists χ in Irr( G) such that χ( g) = 0. In this paper, we consider the set of orders of the vanishing elements of a group G, and we define the prime graph on it, which we denote by Γ( G). Focusing on the class of solvable groups, we prove that Γ( G) has at most two connected components, and we characterize the case when it is disconnected. Moreover, we show that the diameter of Γ( G) is at most 4. Examples are given to round out our understanding of this matter. Among other things, we prove that the bound on the diameter is best possible, and we construct an infinite family of examples showing that there is no universal upper bound on the size of an independent set of Γ( G). [ABSTRACT FROM AUTHOR]

Abstract: Let G be a finite group and p a prime number. We say that an element g in G is a vanishing element of G if there exists an irreducible character χ of G such that . The main result of this paper shows that, if G does not have any vanishing element of p-power order, then G has a normal Sylow p-subgroup. Also, we prove that this result is a generalization of some classical theorems in Character Theory of finite groups. [Copyright &y& Elsevier]

Abstract: In this paper we describe the structure of finite groups whose real-valued nonlinear irreducible characters have all prime degree. The more general situation in which the real-valued irreducible characters of a finite group have all squarefree degree is also considered. [Copyright &y& Elsevier]

Abstract: It is well known that if D is an irreducible complex representation of a finite group G, then every direct summand of the restriction of D to a subgroup H must have degree at least as large as the degree of D divided by the index ; moreover, D is induced from H if and only if the restriction does have a direct summand whose dimension is equal to this quotient. This paper explores the possibility of an analogous result for tensor induction, under the additional assumption that D is faithful, quasi-primitive and not a tensor product (of projective representations of degree greater than 1), and that the Fitting subgroup is not in the centre . The main question is this: if the restriction has a (projective) tensor factor whose degree is the th root of the degree of D, does it follow that D is tensor induced from H? Among other results, examples are given to show that the answer can be negative when the index is 2. An affirmative answer is proved for normal subgroups of odd index, and also for arbitrary subgroups of odd prime index. As might be expected, the key lies in the study of as a symplectic module over a finite prime field; in particular, in exploring the connection between (ordinary) induction and form-induction of such modules. [Copyright &y& Elsevier]

For an irreducible complex character χ of the finite group G , let π (χ) denote the set of prime divisors of the degree χ (1) of χ. Denote then by ρ (G) the union of all the sets π (χ) and by σ (G) the largest value of | π (χ) | , as χ runs in Irr (G). The ρ-σ conjecture , formulated by Bertram Huppert in the 80's, predicts that | ρ (G) | ≤ 3 σ (G) always holds, whereas | ρ (G) | ≤ 2 σ (G) holds if G is solvable; moreover, O. Manz and T.R. Wolf proposed a "strengthened" form of the conjecture in the general case, asking whether | ρ (G) | ≤ 2 σ (G) + 1 is true for every finite group G. In this paper we study the strengthened ρ - σ conjecture for the class of finite groups having a trivial Fitting subgroup : in this context, we prove that the conjecture is true provided σ (G) ≤ 5 , but it is false in general if σ (G) ≥ 6. Instead, we establish that | ρ (G) | ≤ 3 σ (G) − 4 holds for every finite group with a trivial Fitting subgroup and with σ (G) ≥ 6 (this being the right, best possible bound). Also, we improve the up-to-date best bound for the solvable case, showing that we have | ρ (G) | ≤ 3 σ (G) whenever G belongs to one particular class including all the finite solvable groups, and we improve the up-to-date best bound obtained in [18] for the general case. [ABSTRACT FROM AUTHOR]

Let G be a finite group. A conjugacy class of G is said to be vanishing if there exists an irreducible character of G which takes the value 0 on the elements of this class. In this note, we describe the groups whose vanishing classes all have size p for some prime p. [ABSTRACT FROM AUTHOR]

In this paper, we study the number of conjugacy classes of maximal cyclic subgroups of a finite group 퐺, denoted η ⁢ (G) . First we consider the properties of this invariant in relation to direct and semi-direct products, and we characterize the normal subgroups 푁 with η ⁢ (G / N) = η ⁢ (G) . In addition, by applying the classification of finite groups whose nontrivial elements have prime order, we determine the structure of G / ⟨ G − ⟩ , where G − is the set of elements of 퐺 generating non-maximal cyclic subgroups of 퐺. More precisely, we show that G / ⟨ G − ⟩ is either trivial, elementary abelian, a Frobenius group or isomorphic to A 5 . [ABSTRACT FROM AUTHOR]

Let G be a finite group, and let cd ( G ) denote the set of degrees of the irreducible complex characters of G . This paper is a contribution to the study of the degree graph of G , that is, the prime graph built on the set cd ( G ) . Namely, we characterize finite groups whose degree graph has three independent vertices (i.e., three vertices that are pairwise non-adjacent). Our result turns out to be a generalization of several previously-known theorems concerning the structure of the degree graph. [ABSTRACT FROM AUTHOR]

Abstract: Given a finite group G, consider the prime graph built on the set of conjugacy class sizes of G. Denoting by the set of vertices of this graph that are not adjacent to at least one other vertex, we show that the Hall -subgroups of G (which do exist) are metabelian. [Copyright &y& Elsevier]

Abstract: Given a finite group , denote by the simple undirected graph whose vertices are the (distinct) non-central conjugacy class sizes of , and for which two vertices of are adjacent if and only if they are not coprime numbers. In this note we prove that is a -regular graph if and only if it is a complete graph with three vertices, and is a -regular graph if and only if it is a complete graph with four vertices. [Copyright &y& Elsevier]

Abstract: Let G be a finite group, and let denote the prime graph built on the set of conjugacy class sizes of G. In this paper, we consider the situation when has “few complete vertices”, and our aim is to investigate the influence of this property on the group structure of G. More precisely, assuming that there exists at most one vertex of that is adjacent to all the other vertices, we show that G is solvable with Fitting height at most 3 (the bound being the best possible). Moreover, if has no complete vertices, then G is a semidirect product of two abelian groups having coprime orders. Finally, we completely characterize the case when is a regular graph. [Copyright &y& Elsevier]

Abstract: Let G be a finite group, and let denote the set of irreducible complex characters of G. An element x of G is non-vanishing if, for every χ in , we have . We prove that, if x is a non-vanishing element of G and the order of x is coprime to 6, then x lies in the Fitting subgroup of G. [Copyright &y& Elsevier]

Let G be a finite solvable group, and let Δ(G) denote the prime graph built on the set of degrees of the irreducible complex characters of G. A fundamental result by P. P. Pálfy asserts that the complement Δ(G)... of the graph Δ(G) does not contain any cycle of length 3. In this paper we generalize Pálfy’s result, showing that Δ(G)... does not contain any cycle of odd length, whence it is a bipartite graph. As an immediate consequence, the set of vertices of Δ(G) can be covered by two subsets, each inducing a complete subgraph. The latter property yields in turn that if n is the clique number of Δ(G), then Δ(G) has at most 2n vertices. This confirms a conjecture by Z. Akhlaghi and H. P. Tong-Viet, and provides some evidence for the famous ρ-σ conjecture by B. Huppert. [ABSTRACT FROM AUTHOR]

Let G be a finite group. An element g of G is called a vanishing element if there exists an irreducible character χ of G such that χ ( g ) = 0 ; in this case, we say that the conjugacy class of g is a vanishing conjugacy class. In this paper, we discuss some arithmetical properties concerning the sizes of the vanishing conjugacy classes in a finite group. [ABSTRACT FROM AUTHOR]

Given a finite group G, denote by the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of G, and set two vertices of to be adjacent if and only if they are not coprime numbers. In this note we prove that, if is a k-regular graph with k ≥ 1, then is a complete graph with vertices. [ABSTRACT FROM AUTHOR]