Fully invariant subgroup of group of prime power order

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

Definition

A subgroup of a group is termed a fully invariant subgroup of group of prime power order if the whole group is a group of prime power order (i.e., a finite p-group for some prime number p) and the subgroup is a fully invariant subgroup.

Examples

Below are some examples of a proper nontrivial subgroup that satisfy the property fully invariant 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 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
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

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

 Group partSubgroup partQuotient part
Center of direct product of D8 and Z2Direct product of D8 and Z2Klein four-groupKlein four-group
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
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-central Z4 in M16M16Cyclic group:Z4Cyclic group:Z4
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
Q8 in SD16Semidihedral group:SD16Quaternion groupCyclic group:Z2
Z4 in direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z4Cyclic group:Z2

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
Iterated agemo subgroup of group of prime power order
Quotient-iterated omega subgroup of group of prime power order
Commutator-verbal subgroup of group of prime power order

Weaker properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Characteristic subgroup of group of prime power order |FULL LIST, MORE INFO
Finite-p-potentially fully invariant subgroup |FULL LIST, MORE INFO
Finite-p-potentially characteristic subgroup Characteristic subgroup of group of prime power order|FULL LIST, MORE INFO
Normal subgroup of group of prime power order Characteristic subgroup of group of prime power order, Finite-p-potentially fully invariant subgroup, P-automorphism-invariant subgroup of finite p-group|FULL LIST, MORE INFO