# Abelian fully invariant subgroup

This article describes a property that arises as the conjunction of a subgroup property: fully invariant 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 fully invariant subgroup or fully invariant abelian subgroup if $H$ is an abelian group as a group in its own right (or equivalently, is an abelian subgroup of $G$) and is also a fully invariant subgroup (or fully characteristic subgroup) of $G$, i.e., for any endomorphism $\sigma$ of $G$, we have $\sigma(H) \subseteq H$.

## Examples

### Examples based on subgroup-defining functions and series

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
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 semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z2Dihedral group:D8
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

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
fully invariant subgroup of abelian group the whole group is an abelian group and the subgroup is a fully invariant subgroup |FULL LIST, MORE INFO
characteristic direct factor of abelian group the whole group is an abelian group and the subgroup is a characteristic subgroup as well as a direct factor |FULL LIST, MORE INFO
abelian verbal subgroup the subgroup is abelian as well as a verbal subgroup |FULL LIST, MORE INFO
verbal subgroup of abelian group the subgroup is a verbal subgroup and the whole group is an abelian group |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian characteristic subgroup abelian and a characteristic subgroup -- invariant under all automorphisms follows from fully invariant implies characteristic follows from characteristic not implies fully invariant in finite abelian group |FULL LIST, MORE INFO
abelian normal subgroup abelian and a normal subgroup -- invariant under all inner automorphisms (via abelian characteristic, 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 Abelian characteristic subgroup|FULL LIST, MORE INFO
abelian subnormal subgroup abelian and a subnormal subgroup Abelian characteristic subgroup|FULL LIST, MORE INFO