For any (Hausdorff) compact group G, denote by $\mathrm{cp}(G)$ the probability that a randomly chosen pair of elements of G commute. We prove that there exists a finite group H such that $\mathrm{cp}(G)= {\mathrm{cp}(H)}/{|G:F|^2}$ , where F is the FC-centre of G and H is isoclinic to F with $\mathrm{cp}(F)=\mathrm{cp}(H)$ whenever $\mathrm{cp}(G)>0$ . In addition, we prove that a compact group G with $\mathrm{cp}(G)>\tfrac {3}{40}$ is either solvable or isomorphic to $A_5 \times Z(G)$ , where $A_5$ denotes the alternating group of degree five and the centre $Z(G)$ of G contains the identity component of G.

For $G$ a finite non-Abelian group we write $c(G)$ for the probability that two randomly chosen elements commute and $k(G)$ for the largest integer such that any $k(G)$-colouring of $G$ is guaranteed to contain a monochromatic quadruple $(x,y,xy,yx)$ with $xy\neq yx$. We show that $c(G)\rightarrow 0$ if and only if $k(G)\rightarrow \infty$.

In parallel to the classical theory of central extensions of groups, we develop a version for extensions that preserve commutativity. It is shown that the Bogomolov multiplier is a universal object parametrizing such extensions of a given group. Maximal and minimal extensions are inspected, and a connection with commuting probability is explored. Such considerations produce bounds for the exponent and rank of the Bogomolov multiplier.