Monomiality of finite groups with some conditions on conjugacy classes (original) (raw)

Arithmetical conditions for the monomiality of finite groups

Russian Mathematical Surveys, 2009

A finite group is said to be monomial group if all its irreducible characters are induced by linear characters of subgroups. We give some sufficient (for the monomiality of a group) conditions on the conjugacy class sizes. And then we investigate a question on possible analogues for the irreducible character degrees.

Products of conjugacy classes and products of irreducible characters in finite groups

Turkish Journal of Mathematics, 2013

Let G be a finite group. If A and B are two conjugacy classes in G , then AB is a union of conjugacy classes in G and η(AB) denotes the number of distinct conjugacy classes of G contained in AB. If χ and ψ are two complex irreducible characters of G , then χψ is a character of G and again we let η(χψ) be the number of irreducible characters of G appearing as constituents of χψ. In this paper our aim is to study the product of conjugacy classes in a finite group and obtain an upper bound for η in general. Then we study similar results related to the product of two irreducible characters.

Real-Imaginary Conjugacy Classes and Real-Imaginary Irreducible Characters in Finite Groups

Mathematical Notes, 2017

Let G be a finite group. A character χ of G is said to be real-imaginary if its values are real or purely imaginary. A conjugacy class C of a in G is real-imaginary if and only if χ(a) is real or purely imaginary for all irreducible characters χ of G. A finite group G is called real-imaginary if all of its irreducible characters are real-imaginary. In this paper, we describe real-imaginary conjugacy classes and irreducible characters and study some results related to the real-imaginary groups. Moreover, we investigate some connections between the structure of group G and both the set of all the real-imaginary irreducible characters of G and the set of all the real-imaginary conjugacy classes of G.

On the length of the conjugacy classes of finite groups

Journal of Algebra, 1990

Let G be a finite group. The question of how certain arithmetical conditions on the degrees of the irreducible characters of G influence the group structure was studied by several authors. Our purpose here is to impose analogous conditions on the lengths of the conjugacy classes of G and to describe the group structure under these conditions.

Erratum to: Real-Imaginary Conjugacy Classes and Real-Imaginary Irreducible Characters in Finite Groups

Matematicheskie Zametki, 2021

Let G be a finite group. A character χ of G is said to be real-imaginary if its values are real or purely imaginary. A conjugacy class C of a in G is real-imaginary if and only if χ(a) is real or purely imaginary for all irreducible characters χ of G. And a finite group G is called real-imaginary if all of its irreducible characters are real-imaginary. In [1], the author described real-imaginary conjugacy classes and real-imaginary characters. In Theorem 1 of [1], he showed that C is a real-imaginary conjugacy class of G if and only if C 2 = (C −1) 2. In the next example, we show that this statement is not true. Example. Let G be a non-Abelian group of order 21, i.e. G = a, b|a 7 = b 3 = 1, b −1 ab = a 2. We can check easily from the character table of G that the conjugacy class C = {a, a 2 , a 4 } satisfies C 2 = (C −1) 2 , but it is not real-imaginary.

Rational irreducible characters and rational conjugacy classes in finite groups

Transactions of the American Mathematical Society, 2007

We prove that a finite group G G has two rational-valued irreducible characters if and only if it has two rational conjugacy classes, and determine the structure of any such group. Along the way we also prove a conjecture of Gow stating that any finite group of even order has a non-trivial rational-valued irreducible character of odd degree.

On Finite Number of Conjugacy Classes in Groups

Communications in Algebra, 2014

The work is inspired by an article of M. Herzog, P. Longobardi, and M. Maj, who considered groups with a nite number of innite conjugacy classes. Their main results were obtained under assumption that the F C-center is of nite index in the group. We consider here innite groups with a nite number of conjugacy classes of any size (F N CC-groups). Hence the F C-center in our case will be nite, but of innite index in the group. Among results on these groups we give a criterion for a wreath product of F N CC-groups to be an F N CC-group.