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 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:

	Group part	Subgroup part	Quotient part
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group
Center of quaternion group	Quaternion group	Cyclic group:Z2	Klein four-group
Center of special linear group:SL(2,3)	Special linear group:SL(2,3)	Cyclic group:Z2	Alternating group:A4
Center of special linear group:SL(2,5)	Special linear group:SL(2,5)	Cyclic group:Z2	Alternating group:A5
Cyclic maximal subgroup of dihedral group:D8	Dihedral group:D8	Cyclic group:Z4	Cyclic group:Z2
First agemo subgroup of direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z2	Klein four-group
First omega subgroup of direct product of Z4 and Z2	Direct product of Z4 and Z2	Klein four-group	Cyclic group:Z2
Klein four-subgroup of alternating group:A4	Alternating group:A4	Klein four-group	Cyclic group:Z3
Normal Klein four-subgroup of symmetric group:S4	Symmetric group:S4	Klein four-group	Symmetric group:S3

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

	Group part	Subgroup part	Quotient part
Center of M16	M16	Cyclic group:Z4	Klein four-group
Center of central product of D8 and Z4	Central product of D8 and Z4	Cyclic group:Z4	Klein four-group
Center of dihedral group:D16	Dihedral group:D16	Cyclic group:Z2	Dihedral group:D8
Center of direct product of D8 and Z2	Direct product of D8 and Z2	Klein four-group	Klein four-group
Center of nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Klein four-group	Klein four-group
Center of semidihedral group:SD16	Semidihedral group:SD16	Cyclic group:Z2	Dihedral group:D8
Central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Quaternion group
Cyclic maximal subgroup of dihedral group:D16	Dihedral group:D16	Cyclic group:Z8	Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16	Semidihedral group:SD16	Cyclic group:Z8	Cyclic group:Z2
Derived subgroup of M16	M16	Cyclic group:Z2	Direct product of Z4 and Z2
Derived subgroup of dihedral group:D16	Dihedral group:D16	Cyclic group:Z4	Klein four-group
Derived subgroup of nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Direct product of Z4 and Z2
Direct product of Z4 and Z2 in M16	M16	Direct product of Z4 and Z2	Cyclic group:Z2
Klein four-subgroup of M16	M16	Klein four-group	Cyclic group:Z4
Non-central Z4 in M16	M16	Cyclic group:Z4	Cyclic group:Z4
Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Dihedral group:D8

Facts

• 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.
• 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

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

Related group properties

• Group in which every abelian characteristic subgroup is central