I mentioned before the following theorem of Hofmann and Russo, extending earlier work by Levai and Pyber on the profinite case.

Theorem 1 (Hofmann and Russo)If is a compact group of positive commuting probability then the FC-center is an open subgroup of with finite-index center .

(I actually stated this theorem incorrectly previously, asserting the conclusion as well; this is clearly false in general, for instance for .)

Here the *FC-center* of a group is the subgroup of elements with finitely many conjugates. In general a group is called FC if each of its elements has finitely many conjugates, and BFC if its elements have boundedly many conjugates. A theorem of Bernhard Neumann states that a group is BFC if and only if is finite.

I noticed today that one can prove this theorem rather easily by adapting the proof of Peter Neumann’s theorem that a finite group with commuting probability bounded away from is small-by-abelian-by-small. Some parts of the argument below are present in scattered places in the above two papers, but I repeat them for completeness.

*Proof:* Let be the normalised Haar measure of , and suppose that

Let be the set of elements in with at most conjugates. Then is closed, since any element with at least distinct conjugates has a neighbourhood such that for all the points are distinct. Since

we see that for all . This implies that the group generated by is generated in at most steps, i.e., , which implies that is an open BFC subgroup of . Since is an increasing sequence of finite-index subgroups it must terminate with some subgroup , and in fact must be the FC-center of . This proves that is an open BFC subgroup of .

In particular in its own right is a compact group with finite (by the theorem of Bernhard Neumann mentioned at the top of the page). Since the commutator map is a continuous map to a discrete set satisfying there must be a neighbourhood of such that . This implies that is open, hence of finite-index in .

For me, the Hofmann-Russo theorem is a negative result: it states that commuting probability does not extend in an interesting way to the category of compact groups. To be more specific we have the following corollary.

Corollary 2If is a compact group of commuting probability then there is a finite group also of commuting probability .

We need a simple lemma before proving the corollary.

Lemma 3For each there is a finite group of commuting probability .

*Proof:* If is odd then has commuting probability . We can use this formula alone and induction on to define appropriate groups . Take and . If is even take . If take . If take .

An *isoclinism* between two groups and is a pair of isomorphisms and which together respect the commutator map . Clearly isoclinism preserves commuting probability. A basic theorem on isoclinism, due to Hall, is that every group is isoclinic to a *stem group*, a group satisfying . We can now prove the corollary.

*Proof:* Proof of corollary: By the theorem the FC-center of has finite-index, say , and moreover has finite-index center and therefore finite commutator subgroup . Let be a stem group isoclinic to . Then and have the same commuting probability, and is finite since , , and , so we can take .