Abelian normal subgroup

Symbol-free definition
A subgroup of a group is termed an abelian normal subgroup if it is abelian as a group and normal as a subgroup.

Generic examples

 * The trivial subgroup is an abelian normal subgroup.
 * The center, and more generally any central subgroup (i.e., any subgroup contained inside the center) is an abelian normal subgroup.
 * For a nilpotent group, any member of the second half of the lower central series is an abelian normal subgroup.

Related group properties
The group property of not having any nontrivial abelian normal subgroup is equivalent to the property of being Fitting-free i.e. not having any nontrivial nilpotent normal subgroup.

Facts

 * Quotient group acts on abelian normal subgroup: One of the main differences between Abelian normal subgroups and other normal subgroups is that for an Abelian normal subgroup, there is a well-defined action of the quotient group on the subgroup. This is the beginning of group cohomology, which essentially looks at the study of groups that have a given Abelian normal subgroup and a given quotient group, with a specified action of the quotient group on the subgroup.
 * Degree of irreducible representation divides index of abelian normal subgroup: For a finite group, the degrees of irreducible representations over an algebraically closed field of characteristic zero divide the index of any Abelian normal subgroup.
 * Maximal among abelian normal implies self-centralizing in nilpotent and maximal among abelian normal implies self-centralizing in supersolvable: In a group that is nilpotent or supersolvable, any subgroup that is maximal among Abelian normal subgroups contains its own centralizer.

Metaproperties
An abelian normal subgroup of an abelian normal subgroup need not be an abelian normal subgroup.

If $$H$$ is an abelian normal subgroup of $$G$$, $$H$$ is also an abelian normal subgroup in any intermediate subgroup $$K$$. This follows from the fact that normality satisfies the same condition: Normality satisfies intermediate subgroup condition.

The image of an sbelian normal subgroup under a surjective homomorphism is an Abelian normal subgroup of the image. This follows from two facts: Abelianness is quotient-closed, and normality satisfies image condition.

An intersection of a nonempty collection of Abelian normal subgroups is again an Abelian normal subgroup. This follows from two facts: Abelianness is subgroup-closed, and normality is strongly intersection-closed.

A join of Abelian normal subgroups of a group need not be an Abelian normal subgroup. A join of finitely many Abelian normal subgroups, however, is guaranteed to be nilpotent.