Derived length and character degrees of solvable groups

We prove that the derived length of a solvable group is bounded in terms of certain invariants associated to the set of character degrees and improve some of the known bounds. We also bound the derived length of a Sylow p-subgroup of a solvable group by the number of different p-parts of the character degrees of the whole group.

Some problems in number theory that arise from group theory

In this expository paper, we present several open problems in number theory that have arisen while doing research in group theory. These problems are on arithmetical functions or partitions. Solving some of these problems would allow to solve some open problem in group theory.

Extending Brauer's Height Zero Conjecture to Blocks with Nonabelian Defect Groups

We propose a generalization of Brauer?s Height Zero Conjecture that considers positive heights. We give strong evidence supporting one half of the generalization and obtain some partial results regarding the other half.

Groups with a small average number of zeros in the character table

Abstract We classify finite groups with a small average number of zeros in the character table.

Sylow numbers and nilpotent Hall subgroups

Abstract Let π be a set of primes and G a finite group. We characterize the existence of a nilpotent Hall π-subgroup of G in terms of the number of Sylow subgroups for the primes in π.

Multiplicities of fields of values of irreducible characters of finite groups

We show that if f f is the largest multiplicity of the fields of values of the irreducible characters of a finite group G G , then | G | |G| is bounded from above in terms of f f .

On the number of constituents of products of characters

It has been conjectured that if the number of distinct irreducible constituents of the product of two faithful irreducible characters of a finite p-group, for p ≥ 5, is bigger than (p + 1)/2, then it is at least p. We give a counterexample to this conjecture.

The average number of Sylow subgroups of a finite group

We prove that if the average Sylow number (ignoring the Sylow numbers that are one) of a finite group G is ⩽7, then G is solvable.

Landau's theorem and the number of conjugacy classes of zeros of characters

Abstract Motivated by a 2004 conjecture by the author and J. Sangroniz, Y. Yang has recently proved that if G is solvable then the index in G of the 8th term of the ascending Fitting series is bounded in terms of the largest number of zeros in a row in the character table of G. In this note, we prove this result for arbitrary finite groups and propose a stronger form of the 2004 conjecture. We conclude the paper showing some possible ways to prove this strengthened conjecture.

Conjugacy classes, characters and products of elements

Recently, Baumslag and Wiegold proved that a finite group $G$ is nilpotent if and only if $o(xy)=o(x)o(y)$ for every $x,y\in G$ of coprime order. Motivated by this result, we study the groups with the property that $(xy)^G=x^Gy^G$ and those with the property that $\chi(xy)=\chi(x)\chi(y)$ for every complex irreducible character $\chi$ of $G$ and every nontrivial $x, y \in G$ of pairwise coprime order. We also consider several ways of weakening the hypothesis on $x$ and $y$. While the result of Baumslag and Wiegold is completely elementary, some of our arguments here depend on (parts of) the classification of finite simple groups.

ZEROS OF CHARACTERS ON PRIME ORDER ELEMENTS

Suppose that G is a finite group, let χ be a faithful irreducible character of degree a power of p and let P be a Sylow p-subgroup of G. If χ(x) ≠ 0 for all elements of G of order p, then P is cyclic or generalized quaternion. * The research of the first author is supported by a grant of the Basque Government and by the University of the Basque Country UPV 127.310-EB160/98. † The second author is supported by DGICYT.

Nilpotent and abelian Hall subgroups in finite groups

[EN] We give a characterization of the finite groups having nilpotent or abelian Hall pi-subgroups that can easily be verified using the character table.

Orbit sizes, character degrees and Sylow subgroups

Sylow subgroups and the number of conjugacy classes of p-elements

Large orbits ofp-groups on characters and applications to character degrees

We prove that if ap-groupA acts on a solvablep′-groupG then there is a “large” orbit on the ordinary complex irreducible characters ofG. As a consequence of this theorem we obtain results that relate ordinary and Brauer character degrees.

Nonsolvable groups with few character degrees

An answer to a question of Isaacs on character degree graphs

Abstract Let N be a normal subgroup of a finite group G. We consider the graph Γ ( G | N ) whose vertices are the prime divisors of the degrees of the irreducible characters of G whose kernel does not contain N and two vertices are joined by an edge if the product of the two primes divides the degree of some of the characters of G whose kernel does not contain N. We prove that if Γ ( G | N ) is disconnected then G / N is solvable. This proves a strong form of a conjecture of Isaacs.

The average element order and the number of conjugacy classes of finite groups

Abstract Let o ( G ) be the average order of the elements of G, where G is a finite group. We show that there is no polynomial lower bound for o ( G ) in terms of o ( N ) , where N ⊴ G , even when G is a prime-power order group and N is abelian. This gives a negative answer to a question of A. Jaikin-Zapirain.

Nondivisibility among character degrees II: Nonsolvable groups

We say that a finite group G is an NDAD-group (no divisibility among degrees) if for any 1 < a < b in the set of degrees of the complex irreducible characters of G, a does not divide b. In this article, we determine the nonsolvable NDAD-groups. Together with the work of Lewis, Moreto and Wolf (J. Group Theory 8 (2005)), this settles a problem raised by Berkovich and Zhmud’, which asks for a classification of the NDAD-groups.

An answer to two questions of Brewster and Yeh on M-groups

Let χ be a (complex) irreducible character of a finite group. Recall that χ is monomial if there exists a linear character λ ∈ Irr(H), where H is some subgroup of G, such that χ = λG. A group is an M -group if all its irreducible characters are monomial. In 1992, B. Brewster and G. Yeh [1] raised the following two questions. Question A. Let M and N be normal subgroups of a group G. Assume that (|G : M |, |G : N |) = 1 and that M and N are M -groups. Does this imply that G is an M -group? ∗Research supported by the Basque Government, the Spanish Ministerio de Ciencia y Tecnoloǵia and the University of the Basque Country

Character degrees, character codegrees and nilpotence class of p-groups

Du and Lewis raised in 2016 the question of whether the nilpotence class of a p-group is bounded in terms of the number of character codegrees. In 2020, Croome and Lewis, gave a positive answer to ...

Huppert's conjecture for character codegrees

Fixed point spaces, primitive character degrees and conjugacy class sizes

Let G be a finite group that acts on a nonzero finite dimensional vector space V over an arbitrary field. Assume that V is completely reducible as a G-module, and that G fixes no nonzero vector of V. We show that some element g ∈ G has a small fixed-point space in V. Specifically, we prove that we can choose g so that dim C V (g) < (1/p)dim V, where p is the smallest prime divisor of |G|.

On the number of zeros in the columns of the character table of a group

Complex group algebras of finite groups: Brauer's Problem 1

Abstract Brauer's Problem 1 asks the following: What are the possible complex group algebras of finite groups? It seems that with the present knowledge of representation theory it is not possible to settle this question. The goal of this paper is to present a partial solution to this problem. We conjecture that if the complex group algebra of a finite group does not have more than a fixed number m of isomorphic summands, then its dimension is bounded in terms of m . We prove that this is true for every finite group if it is true for the symmetric groups. The problem for symmetric groups reduces to an explicitly stated question in number theory or combinatorics.

Notes on the average number of Sylow subgroups of finite groups

We show that if the average number of (nonnormal) Sylow subgroups of a finite group is less than $${{29} \over 4}$$ then G is solvable or G/F(G) ≌ A5. This generalizes an earlier result by the third author.

Erratum to “Orbit sizes, character degrees and Sylow subgroups” [Adv. Math. 184 (2004) 18–36]

Coprime actions and degrees of primitive inducers of invariant characters

Let a finite group A act coprimely on a finite group G and χ ∈ Irr A(G). Isaacs, Lewis and Navarro proved that if G is nilpotent then the degrees of any two A-primitive characters of A-invariant subgroups of G inducing χ coincide. In this note we aim at extending this result by weakening the hypothesis on G.

A Dual Version of Huppert's  -  Conjecture

Huppert’s ρ-σ conjecture asserts that any finite group has some character degree that is divisible by “many” primes. In this note, we consider a dual version of this problem, and we prove that for any finite group there is some prime that divides “many” character degrees.

BOUNDING THE NUMBER OF IRREDUCIBLE CHARACTER DEGREES OF A FINITE GROUP IN TERMS OF THE LARGEST DEGREE

We conjecture that the number of irreducible character degrees of a finite group is bounded in terms of the number of prime factors (counting multiplicities) of the largest character degree. We prove that this conjecture holds when the largest character degree is prime and when the character degree graph is disconnected.

On the number of conjugacy classes of zeros of characters

Letm be a fixed non-negative integer. In this work we try to answer the following question: What can be said about a (finite) groupG if all of its irreducible (complex) characters vanish on at mostm conjugacy classes? The classical result of Burnside about zeros of characters says thatG is abelian ifm=0, so it is reasonable to expect that the structure ofG will somehow reflect the fact that the irreducible characters vanish on a bounded number of classes. The same question can also be posed under the weaker hypothesis thatsome irreducible character ofG hasm classes of zeros. For nilpotent groups we shall prove that the order is bounded by a function ofm in the first case but only the derive…

HEIGHTS OF CHARACTERS IN BLOCKS OF $p$-SOLVABLE GROUPS

In this paper, it is proved that if $B$ is a Brauer $p$ -block of a $p$ -solvable group, for some odd prime $p$ , then the height of any ordinary character in $B$ is at most $2b$ , where $p^b$ is the largest degree of the irreducible characters of the defect group of $B$ . Some other results that relate the heights of characters with properties of the defect group are obtained.

A graph associated with the $\pi$-character degrees of a group

Let G be a group and $\pi$ be a set of primes. We consider the set ${\rm cd}^{\pi}(G)$ of character degrees of G that are divisible only by primes in $\pi$. In particular, we define $\Gamma^{\pi}(G)$ to be the graph whose vertex set is the set of primes dividing degrees in ${\rm cd}^{\pi}(G)$. There is an edge between p and q if pq divides a degree $a \in {\rm cd}^{\pi}(G)$. We show that if G is $\pi$-solvable, then $\Gamma^{\pi}(G)$ has at most two connected components.

Order of products of elements in finite groups

If G is a finite group, p is a prime, and x∈G, it is an interesting problem to place x in a convenient small (normal) subgroup of G, assuming some knowledge of the order of the products xy, for certain p‐elements y of G.

Transitive permutation groups in which all derangements are involutions

AbstractLet G be a transitive permutation group in which all derangements are involutions. We prove that G is either an elementary abelian 2-group or is a Frobenius group having an elementary abelian 2-group as kernel. We also consider the analogous problem for abstract groups, and we classify groups G with a proper subgroup H such that every element of G not conjugate to an element of H is an involution.

Complex group algebras of finite groups: Brauer’s Problem 1

Brauer’s Problem 1 asks the following: what are the possible complex group algebras of finite groups? It seems that with the present knowledge of representation theory it is not possible to settle this question. The goal of this paper is to announce a partial solution to this problem. We conjecture that if the complex group algebra of a finite group does not have more than a fixed number m m of isomorphic summands, then its dimension is bounded in terms of m m . We prove that this is true for every finite group if it is true for the symmetric groups.

On the number of different prime divisors of element orders

We prove that the number of different prime divisors of the order of a finite group is bounded by a polynomial function of the maximum of the number of different prime divisors of the element orders. This improves a result of J. Zhang.

Homogeneous products of characters

I. M. Isaacs has conjectured (see \cite{isa00}) that if the product of two faithful irreducible characters of a solvable group is irreducible, then the group is cyclic. In this paper we prove a special case of the following conjecture, which generalizes Isaacs conjecture. Suppose that $G$ is solvable and that $\psi,\phi\in\Irr(G)$ are faithful. If $\psi \phi=m\chi$ where $m$ is a positive integer and $\chi \in \Irr(G)$ then $\psi$ and $\phi$ vanish on $G- Z(G)$. In particular we prove that the above conjecture holds for $p$-groups.

A variation on theorems of Jordan and Gluck

Abstract Gluck proved that any finite group G has an abelian subgroup A such that | G : A | is bounded by a polynomial function of the largest degree of the complex irreducible characters of G . This improved on a previous bound of Isaacs and Passman. In this paper, we present a variation of this result that looks at the number of prime factors. All these results, in turn, may be seen as variations on the classical theorem of Jordan on linear groups.

Groups with exactly one irreducible character of degree divisible byp

Let [math] be a prime. We characterize those finite groups which have precisely one irreducible character of degree divisible by [math] .

Field of values of cut groups and k-rational groups

Abstract Motivated by a question of A. Bachle, we prove that if the field of values of any irreducible character of a finite group G is imaginary quadratic or rational, then the field generated by the character table Q ( G ) / Q is an extension of degree bounded in terms of the largest alternating group that appears as a composition factor of G. In order to prove this result, we extend a theorem of J. Tent on quadratic rational solvable groups to nonsolvable groups.