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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 GL n {\displaystyle \operatorname {GL} _{n}} is the subgroup consisting of scalar matrices, i.e. matrices {\displaystyle } with a 11 = ⋯ = a n n {\displaystyle a_{11}=\dots =a_{nn}} and a i j = 0 {\displaystyle a_{ij}=0} for i ≠ j {\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 SL n {\displaystyle \operatorname {SL} _{n}} is semi-simple, for example.
The subgroup of unipotent elements in the radical is called the unipotent radical, it serves to define reductive groups.