Commuting Probability for Approximate Subgroups of a Finite Group

Abstract For subsets X, Y of a finite group G, we write Pr (X, Y) for the probability that two random elements x X and y Y commute. This paper addresses the relation between the structure of an approximate subgroup A G and the probabilities Pr (A, G) and Pr (A, A). The following results are obtained. Theorem 1. 1: Let A be a K-approximate subgroup of a finite group G, and let Pr (A, G) 0. There are two (, K) -bounded positive numbers γ and K0 such that G contains a normal subgroup T and a K0-approximate subgroup B such that |A B| \; max\|A|, |B|\ while the index G: T and the order of the commutator subgroup T, B are (, K) -bounded. Theorem 1. 2: Let A be a K-approximate subgroup of a finite group G, and let Pr (A, A) 0. There are two (, K) -bounded positive numbers γ and s and a subgroup C G such that |C A²| |A| and |C^| s. In particular, A is contained in the union of at most ^-1K² left cosets of the subgroup C. It is also shown that the above results admit approximate converses.