Lucia Sanus

A partition of characters associated to nilpotent subgroups

IfG is a finite solvable group andH is a maximal nilpotent subgroup ofG containingF(G), we show that there is a canonical basisP(G|H) of the space of class functions onG vanishing off anyG-conjugate ofH which consists of characters. ViaP(G|H) it is possible to partition the irreducible characters ofG into “blocks”. These behave like Brauerp-blocks and a Fong theory for them can be developed.

Partial characters with respect to a normal subgroup

Suppose that G is a π-separable group. Let N be a normal π 1 -subgroup of G and let H be a Hall π-subgroup of G . In this paper, we prove that there is a canonical basis of the complex space of the class functions of G which vanish of G -conjugates of HN . When N = 1 and π is the complement of a prime p , these bases are the projective indecomposable characters and set of irreduciblt Brauer charcters of G .

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 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.

Character degree graphs, blocks and normal subgroups

There are several graphs associated with the set of character degrees of a finite group that have been studied. Results on these graphs are often useful when proving results giving structural information of the group from properties of character degrees. The most commonly studied graph is the graph GðGÞ whose vertices are the prime divisors of character degrees of the group G, with two vertices joined by an edge if the product of the primes divides some character degree. This graph was introduced in [7]. However, it is often interesting in character theory to study only certain subsets of the set of character degrees of a group. For instance, the sets of degrees of the members of IrrðGjNÞ 1⁄4 fw A IrrðGÞ jNGKer wg or IrrpðGÞ 1⁄4 fw A IrrðGÞ j wð1Þ is a p-numberg have been widely studied. The graphs associated to these sets of degrees have also been studied. See [2], [3], [8]. The goal of this paper is to introduce two new graphs associated to certain subsets of character degrees and to prove that they share some of the properties of the previously studied graphs. A situation often of interest in character theory is the following. We have a normal subgroup N of a finite group G; y A IrrðNÞ and we want to study the characters of G lying over y. As usual, we write IrrðGjyÞ to denote this set of characters. Our first graph considers these characters. We define the graph GðGjyÞ as follows. Its vertices are the prime divisors of the numbers wð1Þ=yð1Þ, where w A IrrðGjyÞ. We join two vertices by an edge if the product of the two di¤erent primes divides some member of fwð1Þ=yð1Þ j w A IrrðGjyÞg. Our main result shows that the number of connected components of this graph behaves as in the previously studied situations.

CHARACTERS INDUCED FROM FULLY RAMIFIED SUBGROUPS

Suppose that G is a finite π -separable group, let cf(G) be the space of complex class functions of G and let Irr(G) be the set of the irreducible complex characters of G. Let K be an arbitrary Hall π -subgroup of G. If N G, it is proved in [9] that there exists a (unique) basis Pπ (G | N ) of the space vcfπ (G | N ) = {φ ∈ cf(G) | φ(x) = 0 if xπ ′ ∈ G − N } satisfying: (I) for each η ∈ Pπ (G | N ), there is γ ∈ Irr(NK) such that η = γ G ; and (D) for any α ∈ Irr(NK), then αG is a non-negative linear combination of Pπ (G | N ). The basis Pπ (G | N ) defines a natural partition of Irr(G) which behaves like the R. Brauer blocks. We link χ, ψ ∈ Irr(G) if there exists η ∈ Pπ (G | N ) such that