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 ${\displaystyle GL_{n}(K)}$ (for a field K) is the subgroup consisting of the matrices $(a_{ij})$ with ${\displaystyle a_{11}=\dots =a_{nn}}$ and $a_{ij}=0$ for $i\neq j$.

An algebraic group is called semisimple if its radical is trivial, i.e., consists of the identity element only. The group ${\displaystyle SL_{n}(K)}$ is semi-simple, for example.

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

References

• "Radical of a group", Encyclopaedia of Mathematics