Radical of an algebraic group

The radical of an algebraic group is the identity component of its maximal normal solvable subgroup. For example, the radical of the general linear group G L n ( K ) (for a field K) is the subgroup consisting of the matrices ( a i j ) with a 11 = ⋯ = a n n and a i j = 0 for i ≠ j .

An algebraic group is called semisimple if its radical is trivial, i.e., consists of the identity element only. The group S L n ( K ) is semi-simple, for example.

The subgroup of unipotent elements in the radical is called the unipotent radical, it serves to define reductive groups.