Automorphism group is transitive on non-identity elements implies characteristically simple
This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property must also satisfy the second group property
View all group property implications | View all group property non-implications
Characteristically simple group
Further information: characteristically simple group
A characteristically simple group is a group whose automorphism group has no proper nontrivial characteristic subgroup.
Given: A group , such that acts transitively on the set of non-identity elements of . A characteristic subgroup of
To prove: Either is trivial, or
Proof: If is trivial, we are done. Otherwise, there exists a non-identity element . We want to show that .
Pick any element . We want to argue that . Clearly, if is the identity element ; otherwise, by assumption, there exists an automorphism such that . But then, since is characteristic, and , we'd have . This completes the proof.
The automorphism group acting transitively, implies that if we start with any element of the group, its characteristic closure is the whole group -- this is precisely the condition of being characteristically simple.