Exponent of quotient group divides exponent of group

Statement

Suppose $G$ is a group, $H$ is a normal subgroup of $G$, and $G/H$ is the quotient group. Then, the exponent of $G/H$ divides the exponent of $G$.

In particular, if $G$ has finite exponent, so does $G/H$. Conversely, if $G/H$ has infinite exponent, so does $G$.

Related facts

• Exponent of subgroup divides exponent of group
• Exponent of extension group is a multiple of lcm of exponents of normal subgroup and quotient group