Tiling the symmetric group by transpositions

TL;DR

This paper investigates conditions under which the symmetric group $S_n$ can be uniquely tiled by transpositions, extending previous results and proposing conjectures about the impossibility of such tilings for all $n \

Contribution

It introduces a new necessary condition involving partition-transitivity for tilings of $S_n$ by transpositions, generalizing earlier findings.

Findings

Established a new necessary condition for tilings involving partition-transitivity.

Generalized Rothaus and Thompson's result on divisibility conditions.

Conjectured that $S_n$ cannot be tiled by transpositions for any $n \\geq 4$.

Abstract

For nonempty subsets $X$ and $Y$ of a group $G$, we say that $(X,Y)$ is a tiling of $G$ if every element of $G$ can be uniquely expressed as $xy$ for some $x\in X$ and $y\in Y$. In 1966, Rothaus and Thompson studied whether the symmetric group $S_n$ with $n\geq3$ admits a tiling $(T_n,Y)$, where $T_n$ consists of the identity and all the transpositions in $S_n$. They showed that no such tiling exists if $1+n(n-1)/2$ is divisible by a prime number at least $\sqrt{n}+2$. In this paper, we establish a new necessary condition for the existence of such a tiling: the subset $Y$ must be partition-transitive with respect to certain partitions of $n$. This generalizes the result of Rothaus and Thompson, as well as a result of Nomura in 1985. We also study whether $S_n$ can be tiled by the set $T_n^*$ of all the transpositions, which finally leads us to conjecture that neither $T_n$ nor $T_n^*$ tiles $S_n$ for any $n\geq4$.