# Solvable radical

From Groupprops

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup

View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

## 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:

- It is the join of all solvable normal subgroups of the group.
- It is the join of all solvable characteristic subgroups of the group.
- It is the unique largest solvable normal subgroup of the group.
- It is the unique largest solvable characteristic subgroup of the group.

For a finite group , the solvable radical is denoted or .

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.