# Fully invariant implies characteristic

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., fully invariant subgroup) must also satisfy the second subgroup property (i.e., characteristic subgroup)

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## Contents

## Statement

Any fully invariant subgroup (also called fully characteristic subgroup) of a group is a characteristic subgroup.

## Definitions used

### Fully invariant subgroup

`Further information: fully invariant subgroup`

A subgroup of a group is termed fully invariant in if, for every endomorphism of , is contained in .

### Characteristic subgroup

`Further information: characteristic subgroup`

A subgroup of a group is termed characteristic in if, for every automorphism of , is contained in .

## Related facts

- Characteristic not implies fully invariant
- Characteristic not implies fully invariant in finite abelian group
- Characteristic equals fully invariant in odd-order abelian group

## Intermediate properties

*For intermediate notions between characteristic subgroup and fully invariant subgroup, click here*.

### Invariance under restricted classes of endomorphisms

- Strictly characteristic subgroup is a subgroup that is invariant under all surjective endomorphisms. Every fully invariant subgroup is strictly characteristic, and every strictly characteristic subgroup is characteristic.
- Injective endomorphism-invariant subgroup is a subgroup that is invariant under all injective endomorphisms. Every fully invariant subgroup is strictly characteristic, and every strictly characteristic subgroup is characteristic.

## Proof

### Proof idea

The idea behind the proof is that since every automorphism is an endomorphism, invariance under *all* endomorphisms implies invariance under automorphisms.

### Using function restriction expressions

This subgroup property implication can be proved by using function restriction expressions for the subgroup properties

View other implications proved this way |read a survey article on the topic

The property of being fully invariant is the invariance property with respect to endomorphisms. It has the function restriction expression:

Endomorphism Function

The property of being characteristic is the invariance property with respect to automorphisms. It has the function restriction expression:

Automorphism Function

Since the left side of the function restriction expression for characteristicity is stronger while the right sides of both function restriction expressions are equal, the property of being characteristic is weaker than the property of being fully invariant.