# Affine Group

In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.

It is a Lie group if K is the real or complex field or quaternions.

### Other articles related to "affine group, affine, group, groups":

Other Affine Groups - PoincarĂ© Group
... The PoincarĂ© group is the affine group of the Lorentz group. ...
Affine Representation
... An affine representation of a topological (Lie) group G on an affine space A is a continuous (smooth) group homomorphism from G to the automorphism group of A, the affine group Aff(A) ... Similarly, an affine representation of a Lie algebra g on A is a Lie algebra homomorphism from g to the Lie algebra aff(A) of the affine group of A ... An example is the action of the Euclidean group E(n) upon the Euclidean space En ...
Euclidean Group - Overview - Relation To The Affine Group
... The Euclidean group E(n) is a subgroup of the affine group for n dimensions, and in such a way as to respect the semidirect product structure of both groups ... real column vector of size n or by a single square matrix of size n + 1, as explained for the affine group ... we read off from this that Euclidean geometry, the geometry of the Euclidean group of symmetries, is therefore a specialisation of affine geometry ...
Group Scheme - Basic Properties
... over a field, one often can analyze a group scheme by treating it as an extension of group schemes with distinguished properties ... Any group scheme G of finite type is an extension of the connected component of the identity (i.e ... the maximal connected subgroup scheme) by a constant group scheme ...

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