General affine group

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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 GA(n,k) or GA_n(k), is defined as the external semidirect product of the vector space k^n by the group GL(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 GA(V), is defined as the external semidirect product of V by GL(V).