Characteristic subgroup of group of prime power order

This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property imposed on the ambient group: group of prime power order
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

Definition

A characteristic subgroup of group of prime power order is a subgroup of a group where the group is a group of prime power order and the subgroup is a characteristic subgroup.

Examples

Below are some examples of a proper nontrivial subgroup that satisfy the property characteristic subgroup in a group that satisfies the property group of prime power order.

Below are some examples of a proper nontrivial subgroup that does not satisfy the property characteristic subgroup in a group that satisfies the property group of prime power order.

Here is a more systematic look at the examples of order $p^{3}$.

Group	Order	Characteristic subgroups	Non-characteristic subgroups
dihedral group:D8 (subgroup structure)	$8=2^{3}$	center of dihedral group:D8, cyclic maximal subgroup of dihedral group:D8, whole group, trivial subgroup	Klein four-subgroups of dihedral group:D8, non-central subgroups of order 2
quaternion group (subgroup structure)	$8=2^{3}$	center of quaternion group, whole group, trivial subgroup	cyclic maximal subgroups of quaternion group
prime-cube order group:U(3,p) (subgroup structure)	$p^{3}$	center, trivial subgroup, whole group	non-central subgroups of order $p$, subgroups (elementary abelian) of order $p^{2}$
semidirect product of cyclic group of prime-square order and cyclic group of prime order (subgroup structure)	$p^{3}$	center, elementary abelian subgroup of order $p^{2}$, trivial subgroup, whole group	non-central subgroups of order $p$, cyclic subgroups of order $p^{2}$

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Verbal subgroup of group of prime power order			(see also list of examples)
Fully invariant subgroup of group of prime power order			(see also list of examples)
Isomorph-free subgroup of group of prime power order			(see also list of examples)

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Finite-p-potentially characteristic subgroup			(see also list of examples)
Normal subgroup of group of prime power order			(see also list of examples)
Characteristic subgroup of finite group

Metaproperties

Metaproperty name	Satisfied?	Proof
transitive subgroup property	Yes	characteristicity is transitive
strongly intersection-closed subgroup property	Yes	characteristicity is strongly intersection-closed
strongly join-closed subgroup property	Yes	characteristicity is strongly join-closed
intermediate subgroup condition	No	characteristicity does not satisfy intermediate subgroup condition (see examples with group of prime power order)
transfer condition	No	(follows from statement for intermediate subgroup condition)
image condition	No	characteristicity does not satisfy image condition (see examples with group of prime power order)
centralizer-closed subgroup property	Yes	characteristicity is centralizer-closed
commutator-closed subgroup property	Yes	characteristicity is commutator-closed
upper join-closed subgroup property	No	characteristicity is not upper join-closed (See example with group of prime power order)

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Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
In fact, since the property is also true for every group as a subgroup of itself, it is a strongly intersection-closed subgroup property.
ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closed
ABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
In fact, since the property is also true for the trivial subgroup in any group, it is a strongly join-closed subgroup property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

Intermediate subgroup condition

NO: This subgroup property does not satisfy the intermediate subgroup condition: it is possible to have a subgroup satisfying the property in the whole group but not satisfying the property in some intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBGROUP CONDITION: View other subgroup properties not satisfying intermediate subgroup condition| View facts about intermediate subgroup condition

Transfer condition

This subgroup property does not satisfy the transfer condition

Image condition

This subgroup property does not satisfy the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property need not satisfies the property

Centralizer-closedness

This subgroup property is centralizer-closed: the centralizer of any subgroup with this property, in the whole group, again has this property
View other centralizer-closed subgroup properties

Commutator-closedness

This subgroup property is commutator-closed: the commutator of two subgroups each with the property, also has the property.
View other commutator-closed subgroup properties

Effect of property operators

Operator	Meaning	Result of application	Proof
potentially operator	characteristic in some larger group of prime power order	finite-p-potentially characteristic subgroup	(by definition)

The potentially operator

Applying the potentially operator to this property gives: finite-p-potentially characteristic subgroup