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 2 If 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 3 For 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 .