Coset correct means on groups and the probability that two elements commute

Abstract

Amenable groups are those admitting an invariant mean -- a finitely additive probability mean that assigns equal ``weight'' to any two translates of the same set. We introduce coset correct means (CCMs), a class of finitely additive means that, for any subgroup, assigns equal weight to all its cosets, weakening and therefore generalising the notion of an invariant mean. We show that, unlike the case for invariant means, every group admits a CCM and give two constructions -- one via the Ultrafilter Lemma and one via the Hahn--Banach Theorem -- both relying on a Theorem of B. H. Neumann. Using CCMs, we define a degree of commutativity for arbitrary groups, measuring the ``probability'' that two random elements of a group commute. We prove that this degree of commutativity is independent of the choice of CCM and is positive precisely for finite-by-abelian-by-finite groups, recovering and unifying previous characterisations. We also introduce a defect function that quantifies the failure of left invariance for finitely additive means, and define the defect of a group as the infimum of these. We then prove a dichotomy: the defect for a group is either 0 or 1, with 0 characterising amenable groups.