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

Definition

In terms of dimension

Let $n$ be a natural number and $k$ be a field. The general affine group of order $n$ over $k$ , denoted $G A (n, k)$ or $G A_{n} (k)$ , is defined as the external semidirect product of the vector space $k^{n}$ by the group $G L (n, k)$ , acting by linear transformations.

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 $G A (V)$ , is defined as the external semidirect product of $V$ by $G L (V)$ .