Search results for "Conjugacy class"
showing 10 items of 50 documents
The minimal number of characters over a normal p-subgroup
2007
Abstract If N is a normal p-subgroup of a finite group G and θ ∈ Irr ( N ) is a G-invariant irreducible character of N, then the number | Irr ( G | θ ) | of irreducible characters of G over θ is always greater than or equal to the number k p ′ ( G / N ) of conjugacy classes of G / N consisting of p ′ -elements. In this paper, we investigate when there is equality.
Groups with soluble minimax conjugate classes of subgroups
2008
A classical result of Neumann characterizes the groups in which each subgroup has finitely many conjugates only as central-by-finite groups. If $\mathfrak{X}$ is a class of groups, a group $G$ is said to have $\mathfrak{X}$-conjugate classes of subgroups if $G/core_G(N_G(H)) \in \mathfrak{X}$ for each subgroup $H$ of $G$. Here we study groups which have soluble minimax conjugate classes of subgroups, giving a description in terms of $G/Z(G)$. We also characterize $FC$-groups which have soluble minimax conjugate classes of subgroups.
On a graph related to permutability in finite groups
2010
For a finite group G we define the graph $\Gamma(G)$ to be the graph whose vertices are the conjugacy classes of cyclic subgroups of G and two conjugacy classes $\{\mathcal {A}, \mathcal {B}\}$ are joined by an edge if for some $\{A \in \mathcal {A},\, B \in \mathcal {B}\, A\}$ and B permute. We characterise those groups G for which $\Gamma(G)$ is complete.
A note on maximal subgroups and conjugacy classes of finite groups
2021
Given a finite group G, two elements are ≡m-related if they lie in exactly the same maximal subgroups of G. This equivalence relation was introduced by P. J. Cameron, A. Lucchini and C. M. Roney-Do...
The set of conjugacy class sizes of a finite group does not determine its solvability
2014
Abstract We find a pair of groups, one solvable and the other non-solvable, with the same set of conjugacy class sizes.
Rationality and Sylow 2-subgroups
2010
AbstractLet G be a finite group. If G has a cyclic Sylow 2-subgroup, then G has the same number of irreducible rational-valued characters as of rational conjugacy classes. These numbers need not be the same even if G has Klein Sylow 2-subgroups and a normal 2-complement.
Landau's theorem and the number of conjugacy classes of zeros of characters
2021
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.
Character Tables and Sylow Subgroups Revisited
2018
Suppose that G is a finite group. A classical and difficult problem is to determine how much the character table knows about the local structure of G and vice versa.
Q7-branes and their coupling to IIB supergravity
2007
We show how, by making use of a new basis of the IIB supergravity axion-dilaton coset, SL(2,R)/SO(2), 7-branes that belong to different conjugacy classes of the duality group SL(2,R) naturally couple to IIB supergravity with appropriate source terms characterized by an SL(2,R) charge matrix Q. The conjugacy classes are determined by the value of the determinant of Q. The (p,q) 7-branes are the branes in the conjugacy class detQ = 0. The 7-branes in the conjugacy class detQ > 0 are labelled by three numbers (p,q,r) which parameterize the matrix Q and will be called Q7-branes. We construct the full bosonic Wess--Zumino term for the Q7-branes. In order to realize a gauge invariant coupling …
Conjugacy classes, characters and products of elements
2019
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.