General affine group

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Template:Field-parametrized linear algebraic group

Definition

In terms of dimension

Let n be a natural number and K be a field. The general affine group or affine general linear group of degree n over K, denoted GA(n,K), GA_n(K), AGL(n,K), or AGL_n(K), is defined as the external semidirect product of the vector space K^n by the general linear group GL(n,K), acting by linear transformations.

While GA(n,K) cannot be realized as a subgroup of GL(n,K), it can be realized as a subgroup of GL(n+1,K) in a fairly typical way: the vector from K^n is the first n entries of the right column, the matrix from GL(n,K) is the top left n \times n block, there is a 1 in the bottom right corner, and zeroes elsewhere on the bottom row.

In terms of vector spaces

Let V be a K-vector space (which may be finite- or infinite-dimensional). The general affine group of V, denoted GA(V), is defined as the external semidirect product of V by GL(V).