Solution
= Solution
For each $g$, the map
$$
L_g:H\longrightarrow gH,\qquad h\longmapsto gh
$$
is surjective by definition of a <coset> and injective by left cancellation. Its inverse is multiplication by $g^{-1}$. Therefore
$$
\boxed{|gH|=|H|\quad\text{for every }g\in G.}
$$
The same argument with right multiplication proves the analogous result for right <cosets>. This equality of sizes is the counting step in <Lagrange's theorem>.