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]

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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.

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.

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				|FULL LIST, MORE INFO
Normal subgroup of group of prime power order				|FULL LIST, MORE INFO