Radical of an algebraic group

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Definition

The radical of an algebraic group is defined in the following equivalent ways:

  1. It is the connected component of identity of the largest possible closed solvable normal subgroup of the algebraic group.
  2. It is the largest possible connected closed solvable normal subgroup.

Note that the radical differs from a Borel subgroup in that a Borel subgroup is a maximal element among all closed connected solvable (not necessarily normal) subgroups, and it is not necessarily unique (however, it is unique up to conjugacy in an algebraically closed field). The radical of an algebraic group is contained in every Borel subgroup, but the Borel subgroups may be strictly bigger.

There is a corresponding notion for groups without algebraic structure, which we also call the radical (or solvable radical) -- this is simply the largest solvable normal subgroup.