Subgroup of finite index has a left transversal that is also a right transversal

Statement
Suppose $$G$$ is a group and $$H$$ is a subgroup of finite index in $$G$$, i.e., the index of $$H$$ in $$G$$ is finite. Then, there exists a subset $$S$$ of $$G$$ such that $$S$$ is both a left transversal and a fact about::right transversal of $$H$$ in $$G$$. In other words, $$S$$ intersects every left coset of $$H$$ in exactly one element, and it also intersects every right coset of $$H$$ in exactly one element.

Other facts about left and right cosets

 * Left cosets partition a group
 * Left cosets are in bijection via left multiplication
 * Left and right coset spaces are naturally isomorphic

Facts used

 * 1) uses::Subgroup of finite group has a left transversal that is also a right transversal
 * 2) uses::Poincare's theorem

Proof outline
We use fact (2) to quotient out by the normal core, and we use fact (1) to prove the result in the quotient and then pull back to the original group.