Solvable radical

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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 solvable normal subgroups of the group.
2. It is the join of all 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.