General affine group
This term associates to every field, a corresponding group property. In other words, given a field, every group either has the property with respect to that field or does not have the property with respect to that field
This group property is natural number-parametrized, in other words, for every natural number, we get a corresponding group property
In terms of dimension
Let be a natural number and be a field. The general affine group of order over , denoted or , is defined as the external semidirect product of the vector space by the group , acting by linear transformations.
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 .