General affine group
Template:Field-parametrized linear algebraic group
Definition
In terms of dimension
Let be a natural number and
be a field. The general affine group or affine general linear group of degree
over
, denoted
,
,
, or
, is defined as the external semidirect product of the vector space
by the general linear group
, acting by linear transformations.
While cannot be realized as a subgroup of
, it can be realized as a subgroup of
in a fairly typical way: the vector from
is the first
entries of the right column, the matrix from
is the top left
block, there is a
in the bottom right corner, and zeroes elsewhere on the bottom row.
In terms of vector spaces
Let be a
-vector space (which may be finite- or infinite-dimensional). The general affine group of
, denoted
, is defined as the external semidirect product of
by
.