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J. Korean Math. Soc. 2013; 50(1): 161-171

Printed January 1, 2013

https://doi.org/10.4134/JKMS.2013.50.1.161

Copyright © The Korean Mathematical Society.

Some consequences of the equation $[x^{n}, y]=1$ on the structure of a compact group

Ahmad Erfanian, Rashid Rezaei, and Behnaz Tolue

Ferdowsi University of Mashhad, Malayer University, Ferdowsi University of Mashhad

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Given an integer $n\geq1$ and a compact group $G$, we find some restrictions for the probability that two randomly picked elements $x^n$ and $y$ of $G$ commute. In the case $n=1$ this notion was investigated by W.~H.~Gustafson in 1973 and its influence on the structure of the group has been studied in the researches of several authors in last years.

Keywords: $n$-th power central elements, commutativity degree, compact groups

MSC numbers: Primary 20C05, 20P05; Secondary 43A05

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