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