# 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 -group for some prime number ) 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 | 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 |