Affine group

It is the Lie group given by $G L (n) \times R^{n}$ .
We have the product

(A, v) (A_{1}, v_{1}) = (A A_{1}, A v_{1} + v)

It represents affine transformations of $R^{n}$ into itself.
It is a semidirect product

A f f (R^{n}) = R^{n} ⋊ G L (n)

We can see it proceeding like for the Euclidean group.