Nonvanishing elements for Brauer characters

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 is a p-regular element which is p-nonvanishing, then g lies in a normal subgroup of G whose p-length and p'-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 [\u2018Zeros of Brauer characters and linear actions of finite groups\u2019, J. Algebra 340 (2011), 104\u2013113]

Finite groups with real conjugacy classes of prime size

We determine the structure of a finite group G whose noncentral real conjugacy classes have prime size. In particular, we show that G is solvable and that the set of the sizes of its real classes is one of the following: {1},{1, 2}, {1, p}, or {1, 2, p}, where p is an odd prime

On the orders of zeros of irreducible characters

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 χ(g)=0. 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

On the vanishing prime graph of solvable groups

Let G be a finite group, and Irr(G) the set of irreducible complex characters of G. We say that an element g is an element of G is a vanishing element of G if there exists chi in Irr(G) such that chi(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 Gamma(G). Focusing on the class of solvable groups, we prove that Gamma(G) has at most two connected components, and we characterize the case when it is disconnected. Moreover, we show that the diameter of Gamma(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 Gamma(G)

On the vanishing prime graph of finite groups

Let G be a finite group. An element g 08 G is called a vanishing element of G if there exists an irreducible complex character \u3c7 of G such that \u3c7(g) = 0. In this paper we study the vanishing prime graph \u393(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 \u393(G) has at most six connected components

On the character degree graph of finite groups

Given a finite group G, let cd (G) denote the set of degrees of the irreducible complex characters of G. The character degree graph of G is defined as the simple undirected graph whose vertices are the prime divisors of the numbers in cd (G) , two distinct vertices p and q being adjacent if and only if pq divides some number in cd (G). In this paper, we consider the complement of the character degree graph, and we characterize the finite groups for which this complement graph is not bipartite. This extends the analysis of Akhlaghi et al. (Proc Am Math Soc 146:1505\u20131513, 2018), where the solvable case was treated

Finite groups with real-valued irreducible characters of prime degree

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

Incomplete vertices in the prime graph on conjugacy class sizes of finite groups

Given a finite group G, consider the prime graph built on the set of conjugacy class sizes of G. Denoting by \u3c00\u3c00 the set of vertices of this graph that are not adjacent to at least one other vertex, we show that the Hall \u3c00\u3c00-subgroups of G (which do exist) are metabelia

Bounding the number of vertices in the degree graph of a finite group

Let G be a finite group, and let cd(G) denote the set of degrees of the irreducible complex characters of G. The degree graph \u394(G) of G is defined as the simple undirected graph whose vertex set V(G) consists of the prime divisors of the numbers in cd(G), two distinct vertices p and q being adjacent if and only if pq divides some number in cd(G). In this note, we provide an upper bound on the size of V(G) in terms of the clique number \u3c9(G) (i.e., the maximum size of a subset of V(G) inducing a complete subgraph) of \u394(G). Namely, we show that |V(G)| 64max2\u3c9(G)+1,3\u3c9(G) 124. Examples are given in order to show that the bound is best possible. This completes the analysis carried out in [1] where the solvable case was treated, extends the results in [3,4,9], and answers a question posed by the first author and H.P. Tong-Viet in [4]