Solution
= Solution
Every pair factors as $(A,b)=(I,b)(A,0)$, and a <linear map> which is also a translation fixes zero and has linear part $I$, hence is the <identity element>[identity]. Thus $G=NH$ and $N\cap H=\{1\}$. The <conjugation> formula in (c) says that the action needed for reconstruction is simply the usual <matrix> action $A:t\mapsto At$ on the additive <group> $F^n$. Therefore
$$
\boxed{G\cong F^n\rtimes\operatorname{GL}_n(F).}
$$