Emanuele Pacifici

Character degree graphs that are complete graphs

Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We define F(G) to be the graph whose vertex set is cd(G) - {1}, and there is an edge between a and b if (a, b) > 1. We prove that if Γ(G) is a complete graph, then G is a solvable group.

On the regularity of a graph related to conjugacy classes of groups

Given a finite group G , denote by ? ( G ) the simple undirected graph whose vertices are the (distinct) non-central conjugacy class sizes of G , and for which two vertices of ? ( G ) are adjacent if and only if they are not coprime numbers. In this note we prove that ? ( G ) is a 2 -regular graph if and only if it is a complete graph with three vertices, and ? ( G ) is a 3 -regular graph if and only if it is a complete graph with four vertices.

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 ∈ 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 ≤ 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 [3]. Introduction and preliminaries An element g of a finite group G is called a nonvanishing element if χ(g) 6= 0 for every irreducible complex character χ ∈ Irr(G); in other words, g is nonvanishing if the column corresponding to g in the character table of G contains no zero entries. In [7] M. Isaacs, G. Navarro and T. Wolf prove that if G is a finite solvable group and g ∈ G is a nonvanishing element of odd order, then g lies in the Fitting subgroup F(G) of G. Moreover, in [4] it is shown that if G is any finite group and g ∈ G is a nonvanishing element of order coprime to 6, then again g ∈ F(G). In this spirit, given a prime number p, we consider elements corresponding to columns with no zero entries in the table of Brauer characters in characteristic p. We say that a p-regular element g ∈ G is a p-nonvanishing element if φ(g) 6= 0 for every irreducible p-Brauer character φ ∈ IBrp(G). In [3] it is proved that if, for a prime p > 3, all p-regular elements of a finite group G are p-nonvanishing (a condition that implies the solvability of G), then G has p′-length at most 2. Assuming (as we may, since every p-Brauer character of G contains Op(G) in its kernel) that Op(G) = 1, this implies that every p-regular element of G lies in Op′pp′(G). The main theorem of the present paper extends this result, in the case p > 7. Theorem A. Let p be a prime number greater than 3, let G be a finite solvable group with Op(G) = 1, and let g be a p-regular element of G that is p-nonvanishing. Then g lies in Op′pp′(G), unless p ∈ {5, 7} and the order of g is divisible by 2 or 3. Example 4.1 shows that the above statement is “optimal” in some sense. Our approach to Theorem A consists in studying a related problem about linear actions

On tensor factorisation for representations of finite groups

We prove that, given a quasi-primitive complex representation D for a finite group G , the possible ways of decomposing D as an inner tensor product of two projective representations of G are parametrised in terms of the group structure of G . More explicitly, we construct a bijection between the set of such decompositions and a particular interval in the lattice of normal subgroups of G .

On Zeros of Characters of Finite Groups

We survey some results concerning the distribution of zeros in the character table of a finite group and its influence on the structure of the group itself.