On Group bijections $\phi $ with $\phi(B)=A$ and $\forall a\in B, a\phi(a) \notin A$

Abstract

A {\em Wakeford pairing} from $S$ onto $T$ is a bijection $\phi : S \to T$ such that $x\phi(x)\notin T,$ for every $x\in S.$ The number of such pairings will be denoted by $\mu(S,T)$. Let $A$ and $ B$ be finite subsets of a group $G$ with $1\notin B$ and $|A|=|B|.$ Also assume that the order of every element of $B$ is $\ge |B|$. Extending results due to Losonczy and Eliahou-Lecouvey, we show that $\mu(B,A)\neq 0.$ Moreover we show that $\mu(B,A)\ge \min \{\frac{||B|+1}{3},\frac{|B|(q-|B|-1)}{2q-|B|-4}\},$ unless there is $a\in A$ such that $|Aa^{-1}\cap B|=|B|-1$ or $Aa^{-1}$ is a progression. In particular, either $\mu(B,B) \ge \min \{\frac{||B|+1}{3},\frac{|B|(q-|B|-1)}{2q-|B|-4}\},$ or for some $a\in B,$ $Ba^{-1}$ is a progression.