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COMMUTATOR EQUATIONS IN FINITE GROUPS

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

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