# Characteristic not implies derivation-invariant

From Groupprops

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

View a complete list of Lie subring property non-implications | View a complete list of Lie subring property implications

Get more facts about characteristic subring of a Lie ring|Get more facts about derivation-invariant Lie subring

## Statement

A characteristic subring of a Lie ring need not be a derivation-invariant Lie subring.

## Related facts

### Similar facts

### Opposite facts

- Fully invariant subgroup of additive group of Lie ring is derivation-invariant and fully invariant
- Characteristic subgroup of additive group of odd-order Lie ring is derivation-invariant and fully invariant
- Derivation equals endomorphism for Lie ring iff it is abelian
- Inner derivation implies endomorphism for class two Lie ring

## Facts used

## Proof

By fact (1), there exists a finite abelian group with a characteristic subgroup that is not fully invariant in . Consider as an abelian Lie ring, with trivial Lie bracket. Then, the automorphisms of as a Lie ring are the same as the automorphisms as a group, so is a characteristic subring of . Further, the derivations of as a Lie ring are precisely its endomorphisms as a group (because the Leibniz rule condition is vacuous). Thus, since is *not* fully invariant by assumption, is not derivation-invariant.