Characteristic not implies derivation-invariant

From Groupprops
Jump to: navigation, search
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

Facts used

  1. Characteristic not implies fully invariant in finite abelian group

Proof

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