Exponent of extension group divides product of exponents of normal subgroup and quotient group

General statement
Suppose $$G$$ is a group, $$N$$ is a normal subgroup, and $$G/N$$ is the corresponding quotient group. Then, the exponent of $$G$$ divides the product of the exponents of $$N$$ and of $$G/N$$.



Corollary for internal semidirect products
Suppose $$G$$ is a group, $$N$$ is a complemented normal subgroup, and $$H$$ is a permutable complement to $$N$$ in $$G$$. Then, the exponent of $$G$$ divides the product of the exponents of $$N$$ and $$H$$.



Related facts

 * Exponent of subgroup divides exponent of group
 * Exponent of quotient group divides exponent of group
 * Exponent of extension group is a multiple of lcm of exponents of normal subgroup and quotient group
 * Exponent of direct product is lcm of exponents

Facts used

 * 1) uses::Complement to normal subgroup is isomorphic to quotient group

Proof
Fact (1) establishes the corollary for internal semidirect products in terms of the original statement.