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

General statement
Suppose $$G$$ is a group and $$N$$ is a normal subgroup of $$G$$. Then, the exponent of $$G$$ is a multiple of the lcm of the exponents of $$N$$ and of $$G/N$$.

In particular, if $$G$$ has finite exponent, so do $$N$$ and $$G/N$$.



Corollary for semidirect products
Suppose $$G$$ is an internal semidirect product $$N \rtimes H$$, where $$N$$ is a complemented normal subgroup and $$H$$ is a permutable complement to $$N$$ in $$G$$. Then, the exponent of $$G$$ is a multiple of the lcm of the exponents of $$N$$ and $$H$$.

Related facts

 * Exponent of extension group divides product of exponents of normal subgroup and quotient group
 * Exponent of semidirect product may be strictly less than product of exponents
 * Exponent of semidirect product may be strictly greater than lcm of exponents
 * Exponent of direct product is lcm of exponents

Facts used

 * 1) uses::Exponent of subgroup divides exponent of group
 * 2) uses::Exponent of quotient group divides exponent of group
 * 3) uses::Complement to normal subgroup is isomorphic to quotient group

Proof
The proof for the general statement follows directly from Facts (1) and (2).

The statement for semidirect products follows from the general statement if we additionally use Fact (3), but it also follows directly from Fact (1) if we look at both $$N$$ and $$H$$ as subgroups.