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

TL;DR

This paper investigates Wakeford pairings in finite groups, establishing conditions for their existence and providing lower bounds on their number, with implications for group structure and combinatorial properties.

Contribution

The paper extends previous results by proving non-zero counts of Wakeford pairings and deriving explicit lower bounds, considering group elements' orders and progression structures.

Findings

Non-zero Wakeford pairings under certain conditions

Lower bounds on the number of Wakeford pairings

Characterization involving progressions in the group

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.