# 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.

Group partSubgroup partQuotient part
Center of M16M16Cyclic group:Z4Klein four-group
Center of dihedral group:D16Dihedral group:D16Cyclic group:Z2Dihedral group:D8
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2Klein four-group
Center of direct product of D8 and Z2Direct product of D8 and Z2Klein four-groupKlein four-group
Center of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Klein four-groupKlein four-group
Center of quaternion groupQuaternion groupCyclic group:Z2Klein four-group
Center of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z2Dihedral group:D8
Central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Quaternion group
Cyclic maximal subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z8Cyclic group:Z2
Cyclic maximal subgroup of dihedral group:D8Dihedral group:D8Cyclic group:Z4Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z8Cyclic group:Z2
D8 in SD16Semidihedral group:SD16Dihedral group:D8Cyclic 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
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
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 M16M16Klein four-groupCyclic group:Z4
Non-central Z4 in M16M16Cyclic group:Z4Cyclic group:Z4
Q8 in SD16Semidihedral group:SD16Quaternion groupCyclic group:Z2
Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Dihedral group:D8

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.

Group partSubgroup partQuotient part
Cyclic maximal subgroups of quaternion groupQuaternion groupCyclic group:Z4Cyclic group:Z2
D8 in D16Dihedral group:D16Dihedral group:D8Cyclic group:Z2
Klein four-subgroups of dihedral group:D8Dihedral group:D8Klein four-groupCyclic group:Z2
Non-characteristic order two subgroups of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Cyclic group:Z4
Non-normal subgroups of M16M16Cyclic group:Z2
Non-normal subgroups of dihedral group:D8Dihedral group:D8Cyclic group:Z2
Z4 in direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z4Cyclic group:Z2

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
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)