Abstract For any (Hausdorff) compact group G , denote by $\mathrm{cp}(G)$ the probability that a randomly chosen pair of elements commute. We prove there exists finite H such $\mathrm{cp}(G)= {\mathrm{cp}(H)}/{|G:F|^2}$ where F is FC-centre and isoclinic to with $\mathrm{cp}(F)=\mathrm{cp}(H)$ whenever $\mathrm{cp}(G)>0$ . In addition, we $\mathrm{cp}(G)>\tfrac {3}{40}$ either solvable or...