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) need not satisfy 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.