We prove that the commuting probability of a finite ring is no larger than the commuting probabilities of its subrings and quotients, and characterize when equality occurs in such a comparison.

A simple rule of probability revision ensures that the final result of a sequence of probability revisions is undisturbed by an alteration in the temporal order of the learning prompting those revisions. This Uniformity Rule dictates that identical learning be reflected in identical ratios of certain new-to-old odds, and is grounded in the old Bayesian idea that such ratios represent what is le...

Introduction: When G is a finite group, we may endow G×G with the structure of a probability space by assigning the uniform distribution. As was pointed out by W.H. Gustafson [10], the probability that a randomly chosen pair of elements of G commute is then k(G) |G| , where k(G) is the number of conjugacy classes of G. We denote this probability by cp(G). It was also noted in [10] that cp(G) ≤ ...

If H is a subgroup of compact group G, the probability that random element commutes with G denoted by Pr(H,G). Let 〈g〉 stand for monothetic generated an g∈G and let K be G. We prove Pr(〈x〉,G)>0 any x∈K if only has open normal T such K/CK(T) torsion. In particular, x∈G virtually central-by-torsion, is, there G/Z(T) also deduce number corollaries this result.