---
tag: GRCEW24985
kind: definition
name: Affine group
msc: [51N10, 20H15, 15A04]
defines: G6GMSEZDB1
uses: [09XNAPTJQ7, 0JF0TVYGWT, 0FCHE902D7, 0H3WZXCZNF, 087VJM5Q77]
sources:
  - url: https://en.wikipedia.org/wiki/Affine_group
---

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $P$ be an [[affine space|09XNAPTJQ7]] over $V$.
The *affine group* $\operatorname{Aff}(P)$ is the set of all bijective [[affine maps|0JF0TVYGWT]] $P \to P$, with composition as the group operation.
For $P = V$ it is written $\operatorname{Aff}(V)$, and for $V = F^n$ it is written $\operatorname{AGL}(n, F)$.

## Remarks

Example: on $\R^2$, the map $f(x) = Ax + b$ with $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and $b = (1, 0)$ is in $\operatorname{AGL}(2, \R)$, since $\det A = 1 \ne 0$; its inverse is $x \mapsto A^{-1}(x - b)$.
Non-example: $f(x) = Ax + b$ with $A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ is affine but not bijective, so it is not in the affine group.
Each element of $\operatorname{AGL}(n, F)$ has the form $x \mapsto Ax + b$ with $A \in \operatorname{GL}(n, F)$ and $b \in F^n$, in a unique way.
The map to the linear part $A$ is a surjective homomorphism onto the [[general linear group|0FCHE902D7]], whose kernel is the group of [[translations|087VJM5Q77]], isomorphic to $F^n$.
Hence $\operatorname{AGL}(n, F) \cong F^n \rtimes \operatorname{GL}(n, F)$, a [[semidirect product|0H3WZXCZNF]].
It is isomorphic to the group of $(n+1) \times (n+1)$ matrices $\begin{pmatrix} A & b \\ 0 & 1 \end{pmatrix}$ with $A \in \operatorname{GL}(n, F)$.