Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 8 / b / Solution

For a subgroup $N\le H$ and $h\in H$, a left coset is $hN=\{hn:n\in N\}$ and a right coset is $Nh=\{nh:n\in N\}$. A normal subgroup satisfies $hN{h}^{-1}=N$ for every $h\in H$, equivalently $hN=Nh$.

If $N$ is normal, the quotient group is the set $H/N$ of cosets, with multiplicationTo check independence of representatives, replace $h,k$ by $h{n}_1,k{n}_2$. Thenbecause normality puts $k^{-1}{n}_{1}k$ in $N$. Thus the operation is well defined. Associativity follows from that in $H$, the identity is $N$, and the inverse of $hN$ is $h^{-1}N$. This proves the group axioms for $H/N$ and explains the role of coset multiplication is well-defined exactly for normal subgroups.