# Second derived subgroup

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

### Definition with symbols

The **second derived subgroup** of a group , denoted , is defined in the following equivalent ways:

- It is the subgroup generated by all elements of the form , where denotes the commutator and .
- It is the normal closure of the subgroup generated by all elements of the form where denotes the commutator and .
- It is the derived subgroup of the derived subgroup of .
- It is the intersection of all subgroups of for which is a metabelian group, i.e., a solvable group of derived length two.