Left quotient of a subset

Definition with symbols
Given a subset $$S$$ of a group $$G$$, the left quotient of $$S$$, denoted as $$S^{-1}S$$, is defined as the set of all elements that can be written as $$b^{-1}a$$ where $$a$$ and $$b$$ are both in $$S$$.

A subset of a group equals its left quotient if and only if it is a subgroup.