Abelian-quotient subgroup
From Groupprops
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
Contents
Definition
Symbol-free definition
A subgroup of a group is termed an Abelian-quotient subgroup if it satisfies the following equivalent conditions:
- It contains the commutator subgroup
- It is a normal subgroup and the quotient of the whole group by it is Abelian
Definition with symbols
A subgroup of a group <math>G</math> is termed an Abelian-quotient subgroup if it satisfies the following equivalent conditions:
-
where </math>G' = [G,G]</math> is the commutator subgroup of
-
is normal in
and
is an Abelian group
Relation with other properties
Stronger properties
- Elementary Abelian-quotient subgroup
- Cyclic quotient-group
- Abelian-completed normal subgroup
- Cocentral subgroup