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

• For a solvable group, the penultimate member of the derived series (i.e., the last member before reaching the trivial subgroup) is an abelian fully invariant subgroup.
• For a nilpotent group, second half of lower central series of nilpotent group comprises abelian groups: In particular, this means that for a group $G$ of nilpotency class $c$, all the subgroups $\gamma_k(G), k \ge (c + 1)/2$ are abelian characteristic subgroups.

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 nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Klein four-groupKlein four-group
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
Derived subgroup of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Direct product of Z4 and Z2
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 |FULL LIST, MORE INFO
characteristic direct factor of 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