On the commutativity degree of compact groups
In any finite group \(G\), for every \(n\geq 2\) the \(n\)-th commutative degree \(d_n(G)\) 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 this definition to every compact group, which is possible since such a group can be considered as a probability space with respect to the normalized Haar measure. It is proved that if \(G\) and \(H\) are two isoclinic compact groups, then \(d_n(G)=d_n(H)\). Moreover \(d_n(G)\leq (3\cdot 2^n-1)/2^{2n+1}\) for every nonabelian compact group \(G\).
scientific article; zbMATH DE number 5638430