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
- 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.