= Solution
The <quotient group> $G/N$ consists of the <cosets> $gN$, with multiplication
$$
\boxed{(gN)(hN)=(gh)N.}
$$
For well-definedness, replacing representatives by $gn_1$ and $hn_2$ changes their product to $gh(h^{-1}n_1h)n_2$. Since $N$ is a <normal subgroup>, the final two factors lie in $N$, so the product coset does not change.
<Associativity> follows from that in $G$; the <identity element> is the coset $N$, and $(gN)^{-1}=g^{-1}N$. Hence these cosets form a <group>. By <Lagrange's theorem>, every subgroup $H\leq G$ has order dividing $|G|$, and its <subgroup index> is $[G:H]=|G|/|H|$. In particular
$$
\boxed{|G/N|=\frac{|G|}{|N|}.}
$$
Normality is needed for the quotient multiplication, whereas Lagrange’s counting formula holds for every subgroup.
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