After constructing quotient groups, we next ask which familiar properties survive after passing from $G$ to $G/H$. Two important answers are immediate but powerful: a quotient of an abelian group is abelian, and a quotient of a cyclic group is cyclic. These results allow us to recognise many quotient groups without writing a complete Cayley table. The key idea is that each coset $aH$ remembers the behaviour of the representative $a$, but only up to multiplication by elements of $H$. Students often confuse the statement with a converse; a quotient may be abelian even when the original group is not.

Let $(G,\circ)$ be a group and let $H$ be a subgroup of $G$. If $G$ is abelian, then $G/H$ is abelian.