Commuting probability for the Sylow subgroups of a finite group

Abstract: For subsets $X,Y$ of a finite group $G$, let $Pr(X,Y)$ denote the probability that two random elements $x\in X$ and $y\in Y$ commute. Obviously, a finite group $G$ is nilpotent if and only if $Pr(P,Q)=1$ whenever $P$ and $Q$ are Sylow subgroups of $G$ of coprime orders. Suppose that $G$ is a finite group in which for any distinct primes $p,q\in\pi(G)$ there is a Sylow $p$-subgroup $P$ and a Sylow $q$-subgroup $Q$ of $G$ such that $Pr(P,Q) \ge \epsilon$. We show that $F_2(G)$ has $\epsilon$-bounded index in $G$. If $G$ is a finite soluble group in which for any prime $p\in\pi(G)$ there is a Sylow $p$-subgroup $P$ and a Hall $p'$-subgroup $H$ such that $Pr(P,H)\ge \epsilon$, then $F(G)$ has $\epsilon$-bounded index in $G$. Moreover, we establish criteria for nilpotency and solubility of $G$ such as: If for any primes $p,q\in\pi(G)$ the group $G$ has a Sylow $p$-subgroup $P$ and a Sylow $q$-subgroup $Q$ with $Pr(P,Q)>2/3$, then $G$ is nilpotent. If for any primes $p,q\in\pi(G)$ the group $G$ has a Sylow $p$-subgroup $P$ and a Sylow $q$-subgroup $Q$ with $Pr(P,Q)>2/5$, then $G$ is soluble.