Abelian characteristic subgroup
From Groupprops
This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property (itself viewed as a subgroup property): Abelian group
View a complete list of such conjunctions
Contents
Definition
A subgroup of a group
is termed an abelian characteristic subgroup if
is abelian as a group (i.e.,
is an abelian subgroup of
and also,
is a characteristic subgroup of
, i.e.,
is invariant under all automorphisms of
.
Examples
Here are some examples of subgroups in basic/important groups satisfying the property:
Group part | Subgroup part | Quotient part | |
---|---|---|---|
A3 in S3 | Symmetric group:S3 | Cyclic group:Z3 | Cyclic group:Z2 |
Here are some examples of subgroups in relatively less basic/important groups satisfying the property:
Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:
Facts
- Second half of lower central series of nilpotent group comprises abelian groups: In particular, this means that for a group
of nilpotency class
, all the subgroups
are abelian characteristic subgroups.
- The center of any group is an abelian characteristic subgroup.
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
cyclic characteristic subgroup | cyclic group and a characteristic subgroup of the whole group | |FULL LIST, MORE INFO | ||
characteristic central subgroup | central subgroup (i.e., contained in the center) and a characteristic subgroup of the whole group | |FULL LIST, MORE INFO | ||
abelian critical subgroup | |FULL LIST, MORE INFO | |||
characteristic subgroup of center | characteristic subgroup of the center | Characteristic central subgroup|FULL LIST, MORE INFO | ||
characteristic subgroup of abelian group | the whole group is an abelian group and the subgroup is characteristic | Characteristic central subgroup, Characteristic subgroup of center|FULL LIST, MORE INFO | ||
maximal among abelian characteristic subgroups | |FULL LIST, MORE INFO | |||
abelian fully invariant subgroup | abelian and a fully invariant subgroup -- invariant under all endomorphisms | |FULL LIST, MORE INFO |