Size of Quotient Group

Question

For a quotient group G/H (where H is a normal subgroup), is |G/H|=|G|/|H|?

i.e. is the size of the quotient group = (size of G)/(size of H)

Thank you.

Answer 1

Yes, and this follows immediately from Lagrange's theorem.

Answer 2

Let $p:G\rightarrow G/H$, the size of $p^{-1}(y)$ is the size of $H$. To see this, remark that $p^{-1}(1_{G/H})=H$ and if $p(x)=y$, the map $p^{-1}(y)\rightarrow H$ defined by $z\rightarrow x^{-1}z$ is a bijection whose inverse is $z\rightarrow xz$. $G$ is the disjoint union of $\{p^{-1}(y), y\in G/H\}$.