Affine group ($\mathrm{Aff}(V)=V⋊\mathrm{GL}(V)$)

For a finite-dimensional vector space $V$, the affine group consists of invertible affine maps $x\mapsto Ax+b$, where $A$ belongs to the general linear group and $b\in V$. Its product is $(A,b)(A^{\prime },b^{\prime })=(A{A}^{\prime },b+A{b}^{\prime })$, making it the semidirect product of the translation vector space and its general linear group. On the real line the positive-dilation subgroup is the real affine group used in the adjacent article.