COMMUTING PROBABILITY FOR APPROXIMATE SUBGROUPS OF A FINITE GROUP

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) ≥ ε > 0. There are two (ε, K)-bounded positive numbers γ and K0 such that G contains a normal subgroup T and a K0approximate subgroup B such that |A ∩ B| ≥ γ max{|A|, |B|} while the index [G : T] and the order of the commutator subgroup [T, 〈B〉] are (ε, K)-bounded. Theorem 1.2: Let A be a K-approximate subgroup of a finite group G, and let Pr(A, A) ≥ ε > 0. There are two (ε, K)-bounded positive numbers γ and s and a subgroup C ≤ G such that |C ∩ A2| > γ|A| and |C′| ≤ s. In particular, A is contained in the union of at most γ−1K2 left cosets of the subgroup C. It is also shown that the above results admit approximate converses.