Past exam of the options II course of the University of Oxford / 2010 / B / B2 / d / Solution

Every affine transformation factors asso $G=NH$. A translation fixing zero has zero translation vector, so $H\cap N=\{(I,0)\}$. Together with normality from part (c), these are the internal semidirect product conditions. ThusThe reconstruction data are the additive group $F^n$, the linear group ${\mathrm{GL}}_n(F)$, and its standard action $A:c\mapsto Ac$ on translations. In pair notation $(b,A)$, the product is $(b,A)(d,C)=(b+Ad,AC)$, equivalent to the formula in part (b).