Probability that the commutator of two group elements is equal to a given element

In this paper we study the probability that the commutator of two randomly chosen elements in a finite group is equal to a given element of that group. Explicit computations are obtained for groups G which | G ′ | is prime and G ′ ≤ Z ( G ) as well as for groups G which | G ′ | is prime and G ′ ∩ Z...

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Published in:Journal of pure and applied algebra 2008-04, Vol.212 (4), p.727-734
Main Authors: Pournaki, M.R., Sobhani, R.
Format: Article
Language:English
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Summary:In this paper we study the probability that the commutator of two randomly chosen elements in a finite group is equal to a given element of that group. Explicit computations are obtained for groups G which | G ′ | is prime and G ′ ≤ Z ( G ) as well as for groups G which | G ′ | is prime and G ′ ∩ Z ( G ) = 1 . This paper extends results of Rusin [see D.J. Rusin, What is the probability that two elements of a finite group commute? Pacific J. Math. 82 (1) (1979) 237–247].
ISSN:0022-4049
1873-1376
DOI:10.1016/j.jpaa.2007.06.013