Search results for "Compact group"

A note on relative isoclinism classes of compact groups

2009

The probability that $x^m$ and $y^n$ commute in a compact group

2013

In a recent article [K.H. Hofmann and F.G. Russo, The probability that $x$ and $y$ commute in a compact group, Math. Proc. Cambridge Phil. Soc., to appear] we calculated for a compact group $G$ the probability $d(G)$ that two randomly picked elements $x, y\in G$ satisfy $xy=yx$, and we discussed the remarkable consequences on the structure of $G$ which follow from the assumption that $d(G)$ is positive. In this note we consider two natural numbers $m$ and $n$ and the probabilty $d_{m,n}(G)$ that for two randomly selected elements $x, y\in G$ the relation $x^my^n=y^nx^m$ holds. The situation is more complicated whenever $n,m>1$. If $G$ is a compact Lie group and if its identity component $G_…

Henstock type integral in harmonic analysis on zero-dimensional groups

2006

AbstractA Henstock type integral is defined on compact subsets of a locally compact zero-dimensional abelian group. This integral is applied to obtain an inversion formula for the multiplicative integral transform.

Finitary shadows of compact subgroups of $$S(\omega )$$

2020

AbstractLet LF be the lattice of all subgroups of the group $$SF(\omega )$$SF(ω) of all finitary permutations of the set of natural numbers. We consider subgroups of $$SF(\omega )$$SF(ω) of the form $$C\cap SF(\omega )$$C∩SF(ω), where C is a compact subgroup of the group of all permutations. In particular, we study their distribution among elements of LF. We measure this using natural relations of orthogonality and almost containedness. We also study complexity of the corresponding families of compact subgroups of $$S(\omega )$$S(ω).

Character correspondences in blocks with normal defect groups

2014

Abstract In this paper we give an extension of the Glauberman correspondence to certain characters of blocks with normal defect groups.

Representations of Finite Groups

2009

Kurzweil-Henstock type integral on zero-dimensional group and some of its application

2008

A Kurzweil-Henstock type integral on a zero-dimensional abelian group is used to recover by generalized Fourier formulas the coefficients of the series with respect to the characters of such groups, in the compact case, and to obtain an inversion formula for multiplicative integral transforms, in the locally compact case.

The probability that $x$ and $y$ commute in a compact group

2010

We show that a compact group $G$ has finite conjugacy classes, i.e., is an FC-group if and only if its center $Z(G)$ is open if and only if its commutator subgroup $G'$ is finite. Let $d(G)$ denote the Haar measure of the set of all pairs $(x,y)$ in $G \times G$ for which $[x,y] = 1$; this, formally, is the probability that two randomly picked elements commute. We prove that $d(G)$ is always rational and that it is positive if and only if $G$ is an extension of an FC-group by a finite group. This entails that $G$ is abelian by finite. The proofs involve measure theory, transformation groups, Lie theory of arbitrary compact groups, and representation theory of compact groups. Examples and re…

Finding Invariants of Group Actions on Function Spaces, a General Methodology from Non-Abelian Harmonic Analysis

2008

In this paper, we describe a general method using the abstract non-Abelian Fourier transform to construct “rich” invariants of group actions on functional spaces.

A probabilistic meaning of certain quasinormal subgroups

2007

The role of the cyclic quasinormal subgroups has been recently described in groups both finite and infinite by S.Stonehewer and G.Zacher. This role can be better analyzed in the class of compact groups, obtaining restrictions for the probability that two randomly chosen elements commute. Mathematcs Subject Classification: 20D60, 20P05, 20D08