Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-26T11:45:50.034Z Has data issue: false hasContentIssue false
  • English
  • Français

COMMUTATOR EQUATIONS IN FINITE GROUPS

Part of: Special aspects of infinite or finite groups Probabilistic methods in group theory Representation theory of groups

Published online by Cambridge University Press:  03 June 2021

KANTO IRIMOTO  and
ENRIQUE TORRES-GIESE
KANTO IRIMOTO
Affiliation:
Trinity Western University, LangleyBC, V2Y 1Y1, Canada e-mails: kanto.irimoto@twu.ca, enrique.torresgiese@twu.ca
ENRIQUE TORRES-GIESE
Affiliation:
Trinity Western University, LangleyBC, V2Y 1Y1, Canada e-mails: kanto.irimoto@twu.ca, enrique.torresgiese@twu.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem of finding the number of ordered commuting tuples of elements in a finite group is equivalent to finding the size of the solution set of the system of equations determined by the commutator relations that impose commutativity among any pair of elements from an ordered tuple. We consider this type of systems for the case of ordered triples and express the size of the solution set in terms of the irreducible characters of the group. The obtained formulas are natural extensions of Frobenius’ character formula that calculates the number of ways a group element is a commutator of an ordered pair of elements in a finite group. We discuss how our formulas can be used to study the probability distributions afforded by these systems of equations, and we show explicit calculations for dihedral groups.

MSC classification

Primary: 20P05: Probabilistic methods in group theory
Secondary: 20C15: Ordinary representations and characters 20F12: Commutator calculus
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 2 , May 2022 , pp. 320 - 335
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adem, A., Cohen, F. and Torres-Giese, E., Commuting elements, simplicial spaces and filtrations of classifying spaces, Math. Proc. Cambridge Philos. Soc. 152(1) (2012), 91–114.CrossRefGoogle Scholar
Adem, A. and Gómez, J. M., A classifying space for commutativity in Lie groups, Algebr. Geom. Topol. 15(1) (2015), 493–535.CrossRefGoogle Scholar
Deaconescu, M. and Walls, G., Remarks on commutators in finite groups, J. Reine Angew. Math. 732 (2017), 247–253.CrossRefGoogle Scholar
Garion, S. and Shalev, A., Commutator maps, measure preservation, and T-systems, Trans. Amer. Math. Soc. 361(9) (2009), 4631–4651.CrossRefGoogle Scholar
Guralnick, R. M. and Robinson, G. R., On the commuting probability in finite groups, J. Algebra 300(2) (2006), 509–528.CrossRefGoogle Scholar
Gustafson, W. H., What is the probability that two group elements commute?, Amer. Math. Monthly 80 (1973), 1031–1034.CrossRefGoogle Scholar
Isaacs, I. M., Character theory of finite groups , Pure and Applied Mathematics, vol. 69 (Academic Press, New York-London, 1976).Google Scholar
Lescot, P., Isoclinism classes and commutativity degrees of finite groups, J. Algebra 177(3) (1995), 847–869.CrossRefGoogle Scholar
Liebeck, M., O’Brien, E. A., Shalev, A. and Tiep, P. H., The Ore conjecture, J. Eur. Math. Soc. 12(4) (2010), 939–1008.CrossRefGoogle Scholar
James, G. and Liebeck, M., Representations and characters of groups, 2nd edition (Cambridge University Press, New York, 2001).CrossRefGoogle Scholar
Pournaki, M. R. and Sobhani, R., Probability that the commutator of two group elements is equal to a given element, J. Pure Appl. Algebra 212(4) (2008), 727–734.CrossRefGoogle Scholar
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.11.0; 2020, https://www.gap-system.org.Google Scholar
Torres-Giese, E., Higher commutativity and nilpotency in finite groups, Bull. Lond. Math. Soc. 44(6) (2012), 1259–1273.CrossRefGoogle Scholar