Commuting Probability for Approximate Subgroups of a Finite Group

Abstract

Abstract For subsets $X,Y$ of a finite group G, we write $\mathrm{Pr} (X,Y)$ for the probability that two random elements $x\in X$ and $y\in Y$ commute. This paper addresses the relation between the structure of an approximate subgroup $A\subseteq G$ and the probabilities $\mathrm{Pr} (A,G)$ and $\mathrm{Pr} (A,A)$. The following results are obtained. Theorem 1.1: Let A be a K-approximate subgroup of a finite group G, and let $\mathrm{Pr} (A,G)\geq\epsilon\gt0$. There are two $(\epsilon, K)$-bounded positive numbers γ and K0 such that G contains a normal subgroup T and a K0-approximate subgroup B such that $|A\cap B|\ge \gamma\; {\mathrm{max}}\{|A|,|B|\}$ while the index $[G:T]$ and the order of the commutator subgroup $[T,\langle B \rangle]$ are $(\epsilon, K)$-bounded. Theorem 1.2: Let A be a K-approximate subgroup of a finite group G, and let $\mathrm{Pr} (A,A)\geq\epsilon\gt0$. There are two $(\epsilon, K)$-bounded positive numbers γ and s and a subgroup $C\leq G$ such that $|C \cap A^2| \gt \gamma |A|$ and $|C^{\prime}|\leq s$. In particular, A is contained in the union of at most $\gamma^{-1}K^2$ left cosets of the subgroup C. It is also shown that the above results admit approximate converses.