Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 4 / 2 / i / Solution

The map between left-coset setsis well defined and injective: equality of the images is equivalent to $s_2^{-1}{s}_1\in H$, and this element already lies in $S$. These are coset sets, not asserted quotient groups, since $H$ need not be normal. Thus the index of a subgroup satisfies