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

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

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

See also

• Reductive group
• Unipotent group

References

• "Radical of a group", Encyclopaedia of Mathematics