Solvable radical

Definition
The solvable radical or radical or solvable core of a finite group, or more generally, for a slender group, is defined in the following equivalent ways:


 * 1) It is the join of all defining ingredient::solvable normal subgroups of the group.
 * 2) It is the join of all defining ingredient::solvable characteristic subgroups of the group.
 * 3) It is the unique largest solvable normal subgroup of the group.
 * 4) It is the unique largest solvable characteristic subgroup of the group.

For a finite group $$G$$, the solvable radical is denoted $$\operatorname{Rad}(G)$$ or $$\mathcal{O}_\infty(G)$$.

For an arbitrary group, the join of all solvable normal subgroups need not be solvable, hence the solvable radical may not be well-defined. However, if the join is solvable, then it satisfies all the conditions (1) - (4) and is termed the solvable radical.

There is a corresponding notion for algebraic groups that is extremely important. For more, see radical of an algebraic group.