Reductive group - Definition and Overview

In mathematics, a reductive group is an algebraic group G such that the unipotent radical of the identity component of G is trivial. Any semisimple algebraic group and any algebraic torus is reductive, as is any general linear group.

The name comes from the complete reducibility of linear representations of such a group, which is a property in fact holding over fields of characteristic zero. Haboush's theorem shows that a certain rather weaker property holds for reductive groups in the general case.