Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 6 / a / ii / Solution

For $W=aH$ and $W^{\prime }=bH$, defineIf $w=ah$ then $f(w)=bh\in bH$, so this function is well defined. Its inverse function is $w^{\prime }\mapsto a{b}^{-1}{w}^{\prime }$, since successive left translation in a group by $b{a}^{-1}$ and $a{b}^{-1}$ cancels. ThusIn particular every left coset has the same cardinality as $H$.