Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 8 / b / Solution

For a subgroup $H\le G$ and $g\in G$, the left and right cosets are $gH=\{gh:h\in H\}$ and $Hg=\{hg:h\in H\}$. A normal subgroup satisfies $gH{g}^{-1}=H$ for every $g$. Multiplying this set equality on the right by $g$ givesConversely this equality implies $gH{g}^{-1}=H$, so it is an equivalent definition of normality.