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

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