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[EN] We obtain some structural properties of a factorised group G = AB, given that the conjugacy class sizes of certain elements in A boolean OR B are not divisible by p(2), for some prime p. The case when G = AB is a mutually permutable product is especially considered., The first author is supported by Proyecto Prometeo II/2015/011, Generalitat Valenciana (Spain), and the second author is supported by Proyecto MTM2014-54707-C3-1-P, Ministerio de Economia, Industria y Competitividad (Spain). The results in this paper are part of the third author's Ph.D. thesis, and he acknowledges the predoctoral grant ACIF/2016/170, Generalitat Valenciana (Spain).

Let G be a finite group. An element g of G is called a vanishing element if there exists an irreducible character χ of G such that χ ( g ) = 0 ; in this case, we say that the conjugacy class of g is a vanishing conjugacy class. In this paper, we discuss some arithmetical properties concerning the sizes of the vanishing conjugacy classes in a finite group.

A lower bound on the number of elements outside a closed subset of a C-basis of a commutative C-algebra with dual a table algebra is derived, as is an equivalent condition for when the lower bound is met. As corollaries, lower bounds are obtained on the number of primitive idempotent matrices of rank greater than 1 in the adjacency algebra of a commutative, imprimitive association scheme; and, for a given normal subgroup N of a finite group G, on the number of irreducible characters of G whose kernels do not contain N, and on the number of conjugacy classes of G not contained in N. Also found are equivalent conditions for when these lower bounds are met.

This paper develops methods to describe the conjugacy classes of GL(n,R) on Rn×n for a serial ring R of length two. The main result is a reduction to a computation in the matrix algebra over the residue class field of R, which in some cases can be done theoretically without any actual computation.

In this paper, we investigate the irreducible characters and conjugacy classes of a metacyclic p-group where p is an odd prime number. We show that, if | G ′ | = p n , then G has precisely | G : G ′ | ( p − 1 ) / p k + 1 irreducible characters of degree p k , for k = 1 , … , n . We also show that G contains precisely | Z ( G ) | ( p k − p k − 2 ) conjugacy classes of length p k , for k = 1 , … , n . Moreover, we have | G | = | Z ( G ) | ⋅ | G ′ | 2 . We also investigate the concept of a centrally large subgroup, as introduced by Glauberman. In addition, we show that these results are, in general, false for p = 2 . Furthermore, we consider p-groups of small rank, for p > 2 .

In this paper we prove the following conjecture of G. Navarro: if G is a finite 2-group with exactly 5 rational conjugacy classes, then G is dihedral, semidihedral or generalized quaternion. We also characterize the 2-groups with 4 rational classes.

Two theorems on the decomposition of products in nilpotent commutative table algebras are established. This generalizes several results of Adan-Bante on the decomposition of products of irreducible characters and of conjugacy classes of finite p -groups, and extends these results to fusion rule algebras and association schemes.

We prove that in a finite group of odd order, the number of irreducible quadratic characters is the number of quadratic conjugacy classes.

The frame of a group is the poset of conjugacy classes of all its proper subgroups. In this paper we will prove that a finite group is solvable if and only if every collection of maximal elements of its frame has a well-defined meet and the poset consisting of all such meets (including the meet of the empty set) is a modular lattice.

We give a definition of weakly sofic groups (w-sofic groups). Our definition is a rather natural extension of the definition of sofic groups where instead of the Hamming metric on symmetric groups we use general bi-invariant metrics on finite groups. The existence of non-w-sofic groups is equivalent to some conjecture about profinite topology on free groups.

Let G be a non-Abelian finite group, Γ ( G ) the attached graph related to its conjugacy classes. The aim of this paper is to prove that the symmetric group S 3 , the dihedral group D 5 , the three pairwise non-isomorphic non-Abelian groups of order 12, and the non-Abelian group T 21 of order 21, is the complete list of all G such that Γ ( G ) contains no triangles.

In earlier work, the first author outlined an algorithm for calculating a parametrization of the conjugacy classes in a Sylow $p$-subgroup $U(q)$ of a finite Chevalley group $G(q)$, valid when $q$ is a power of a good prime for $G(q)$. In this paper we develop this algorithm and discuss an implementation in the computer algebra language {\sf GAP}. Using the resulting computer program we are able to calculate the parametrization of the conjugacy classes in $U(q)$, when $G(q)$ is of rank at most 6. In these cases, we observe that the number of conjugacy classes of $U(q)$ is given by a polynomial in $q$ with integer coefficients., 14 pages: Significant revisions and explanations expanded

We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a ``matrix problem''. Such problems involve finding normal forms for matrices under a specified set of row and column operations. We solve the relevant matrix problem in small dimensional cases. This gives us all conjugacy classes in maximal parabolic subgroups over a perfect field when one of the two blocks has dimension less than 6. In particular, this includes every maximal parabolic subgroup of GL_n(k) for n < 12 and k a perfect field. If our field is finite of size q, we also show that the number of conjugacy classes, and so the number of characters, of these groups is a polynomial in $q$ with integral coefficients., 23 pages, 6 figures. See also http://zaphod.uchicago.edu/~murray/research/index.html . Submitted to Journal of Algebra

Let G be a finite solvable group. We assume that the set of conjugacy class sizes of G is {1, m, n, m k} with m and n coprime positive integers greater than 1 and k a divisor of n. Then we obtain several properties on the structure of G. © 2007 Elsevier Inc. All rights reserved.

Let G be a finite p-solvable group. We prove that if the set of conjugacy class sizes of all p ′ -elements of G is { 1 , m , p a , m p a } , where m is a positive integer not divisible by p, then the p-complements of G are nilpotent and m is a prime power. This result partially extends a theorem for ordinary classes which asserts that if the set of conjugacy class sizes of a finite group G is exactly { 1 , m , n , m n } and ( m , n ) = 1 , then G is nilpotent.