# 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 $G$, denoted $\! G''$, is defined in the following equivalent ways:

1. It is the subgroup generated by all elements of the form $[[x,y],[z,w]]$, where $[,]$ denotes the commutator and $x,y,z,w \in G$.
2. It is the normal closure of the subgroup generated by all elements of the form $[[x,y],[z,w]]$ where $[,]$ denotes the commutator and $x,y,z,w \in G$.
3. It is the derived subgroup of the derived subgroup of $G$.
4. It is the intersection of all subgroups $H$ of $G$ for which $G/H$ is a metabelian group, i.e., a solvable group of derived length two.