COMMUTING PROBABILITY OF COMPACT GROUPS

Abstract

For any (Hausdorff) compact group G, denote by cp(G) the probability that a randomly chosen pair of elements of G commute. We prove that there exists a finite group H such that cp(G) = cp(H)/|G : F|2, where F is the FC-centre of G and H is isoclinic to F with cp(F) = cp(H) whenever cp(G) > 0. In addition, we prove that a compact group G with cp(G) > 3/40 is either solvable or isomorphic to A5 × Z(G), where A5 denotes the alternating group of degree five and the centre Z(G) of G contains the identity component of G. © 2021 Australian Mathematical Publishing Association Inc.