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 [math]\displaystyle{ \operatorname{GL}_n(K) }[/math] (for a field K) is the subgroup consisting of scalar matrices, i.e. matrices [math]\displaystyle{ (a_{ij}) }[/math] with [math]\displaystyle{ a_{11} = \dots = a_{nn} }[/math] and [math]\displaystyle{ a_{ij}=0 }[/math] for [math]\displaystyle{ i \ne j }[/math].

An algebraic group is called semisimple if its radical is trivial, i.e., consists of the identity element only. The group [math]\displaystyle{ \operatorname{SL}_n(K) }[/math] is semi-simple, for example.

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

See also

• Reductive group
• Unipotent group

References

• "Radical of a group", Encyclopaedia of Mathematics