# Characteristic not implies injective endomorphism-invariant

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) neednotsatisfy the second subgroup property (i.e., injective endomorphism-invariant subgroup)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

Get more facts about characteristic subgroup|Get more facts about injective endomorphism-invariant subgroup

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## Contents

## Statement

### Statement with symbols

It is possible to have a group with a characteristic subgroup that is not injective endomorphism-invariant: in other words, every automorphism of sends to itself, but every injective endomorphism of does *not* send to itself.

## Facts used

- Finitary symmetric group is characteristic in symmetric group
- Finitary symmetric group is not injective endomorphism-invariant in symmetric group
- Center is characteristic
- Center not is injective endomorphism-invariant

## Proof

### Example of the finitary symmetric group

Let be an infinite set. Let be the symmetric group on , and let be the subgroup comprising finitary permutations. Then, is characteristic in (fact (1)) but is not I-characteristic in (fact (2)).

### Example of the center

Facts (3) and (4) give another kind of example: the center of a group is always characteristic, but need not be injective endomorphism-invariant.