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

Abstract

A {\em Wakeford pairing} from $S$ onto $T$ is a bijection $\varphi :S\to T$ such that $x\varphi (x)\notin T,$ 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 $\mu (B,A)\ne 0.$ 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 $|A{a}^{-1}\cap B|=|B|-1$ or $A{a}^{-1}$ 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 $a\in B,$ $B{a}^{-1}$ is a progression.