Order of extension group is product of order of normal subgroup and quotient group

Statement
Suppose $$G$$ is a group, $$N$$ is a normal subgroup, and $$G/N$$ is the corresponding quotient group. Then, we have the following relation between the orders of $$G,N,G/N$$:

$$|G| = |N||G/N|$$

In other words, the order of the extension group for a group extension is the product of the orders of the normal subgroup and the quotient group.

Note that:


 * The order of $$G$$ is finite if and only if both the order of $$N$$ and the order of $$G/N$$ are finite, and in this case the statement is interpreted using multiplication of natural numbers.
 * If any of the orders is infinite, the statement is interpreted in terms of cardinal multiplication.

Related facts

 * Lagrange's theorem
 * Order of quotient group divides order of group
 * Order of element divides order of group

Facts used

 * 1) uses::Lagrange's theorem

Proof
The proof follows directly from Fact (1).