Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / B / B2 / c

The map $A\mapsto (A,0)$ is an injective group homomorphism, so its image is the required copy of ${\mathrm{GL}}_n(F)$. Translations are the subgroup $N=\{(I,b):b\in F^n\}$, isomorphic to the additive group of $F^n$. For any affine map,Thus conjugation preserves translations, proving $N⊴G$.