Normal Subgroups in Group Theory

Solution 1:

Your first statement true. The reason is because each element of $G/H$ is a coset of $H$ of the form $g+H$ and so each element consists of $|H|$ elements of $G$. AS every element of $G$ is in exactly one coset, we have $|G/H| = |G| / |H|$, which gives your formula.

As for your second statement, it is not entirely clear what you are asking. Note that normality is a sufficient and necessary condition for the quotient group to be well-defined - in other words, for operations in the quotient to not depend on which representatives from $H$ are used. So normal subgroups and quotient groups are innately linked.