Second derived subgroup
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
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.