On the commutativity degree of compact groups

In any finite group G , the commutativity degree of G (denoted by d ( G )) is the probability that two randomly chosen elements of G commute. More generally, for every n ≥ 2 the n th commutativity degree (denoted by d n ( G )) is the probability that a randomly chosen ordered ( n + 1)-tuple of the group elements is mutually commuting. The aim of this paper is to generalize the definition of d ( G ) and d n ( G ) to every compact group G (infinite and even uncountable). We shall state some results concerning compact groups and we will extend some results in Erfanian et al. (Comm. Algebra 35 (2007), 4183–4197) and Lescot (J. Algebra 177 (1995), 847–869).