# Abelian characteristic subgroup

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

## Definition

A subgroup $H$ of a group $G$ is termed an abelian characteristic subgroup if $H$ is abelian as a group (i.e., $H$ is an abelian subgroup of $G$ and also, $H$ is a characteristic subgroup of $G$, i.e., $H$ is invariant under all automorphisms of $G$.

## Examples

Here are some examples of subgroups in basic/important groups satisfying the property:

Group partSubgroup partQuotient part
A3 in S3Symmetric group:S3Cyclic group:Z3Cyclic group:Z2

Here are some examples of subgroups in relatively less basic/important groups satisfying the property:

Group partSubgroup partQuotient part
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2Klein four-group
Center of quaternion groupQuaternion groupCyclic group:Z2Klein four-group
Center of special linear group:SL(2,3)Special linear group:SL(2,3)Cyclic group:Z2Alternating group:A4
Center of special linear group:SL(2,5)Special linear group:SL(2,5)Cyclic group:Z2Alternating group:A5
Cyclic maximal subgroup of dihedral group:D8Dihedral group:D8Cyclic group:Z4Cyclic group:Z2
First agemo subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Klein four-group
First omega subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Klein four-groupCyclic group:Z2
Klein four-subgroup of alternating group:A4Alternating group:A4Klein four-groupCyclic group:Z3
Normal Klein four-subgroup of symmetric group:S4Symmetric group:S4Klein four-groupSymmetric group:S3

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

Group partSubgroup partQuotient part
Center of M16M16Cyclic group:Z4Klein four-group
Center of dihedral group:D16Dihedral group:D16Cyclic group:Z2Dihedral group:D8
Center of direct product of D8 and Z2Direct product of D8 and Z2Klein four-groupKlein four-group
Center of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z2Dihedral group:D8
Cyclic maximal subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z8Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z8Cyclic group:Z2
Derived subgroup of M16M16Cyclic group:Z2Direct product of Z4 and Z2
Derived subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z4Klein four-group
Direct product of Z4 and Z2 in M16M16Direct product of Z4 and Z2Cyclic group:Z2
Klein four-subgroup of M16M16Klein four-groupCyclic group:Z4
Non-central Z4 in M16M16Cyclic group:Z4Cyclic group:Z4

## 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

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian normal subgroup abelian and a normal subgroup -- invariant under all inner automorphisms follows from characteristic implies normal follows from normal not implies characteristic in the collection of all groups satisfying a nontrivial finite direct product-closed group property |FULL LIST, MORE INFO
abelian subnormal subgroup abelian and a subnormal subgroup |FULL LIST, MORE INFO
class two characteristic subgroup characteristic subgroup that is also a group of nilpotency class two |FULL LIST, MORE INFO
nilpotent characteristic subgroup characteristic subgroup that is also a nilpotent group follows from abelian implies nilpotent follows from nilpotent not implies abelian Class two characteristic subgroup|FULL LIST, MORE INFO
solvable characteristic subgroup characteristic subgroup that is also a solvable group (via nilpotent) (via nilpotent) Nilpotent characteristic subgroup|FULL LIST, MORE INFO