# Direct factor not implies characteristic

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., direct factor) neednotsatisfy the second subgroup property (i.e., characteristic subgroup)

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## Contents

## Statement

A direct factor of a group need not be a characteristic subgroup.

## Related facts

### Similar facts for subgroup properties

- Normal not implies characteristic
- Normal not implies direct factor
- Characteristic not implies direct factor

### Analogues in Lie rings

- Direct factor not implies derivation-invariant
- Direct factor not implies characteristic in Lie rings

## Proof

### Example of a direct product

Let be any nontrivial group. Then consider , viz., the external direct product of with itself. The subgroups and are direct factors of , and are hence both normal in . Note also that they are distinct, since is nontrivial.

However, the exchange automorphism:

exchanges the subgroups and . Thus, neither nor is invariant under all the automorphisms, so neither is characteristic. Thus, and are both direct factors of that are not characteristic.