Right coset transversal

A right coset transversal for a subgroup $A\le G$ chooses one representative $r$ for each right coset $Ar$. Every $g\in G$ then has a unique expression $g=ar$ with $a\in A$ and $r$ in the transversal. Choosing $1$ as the representative of $A$ makes membership in $A$ equivalent to having representative $1$.