Automorphism group is transitive on nonidentity 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
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Contents
Statement
Verbal statement
If a group has the property that its automorphism group acts transitively on nonidentity elements, then the group is characteristically simple.
Definitions used
Characteristically simple group
Further information: characteristically simple group
A characteristically simple group is a group whose automorphism group has no proper nontrivial characteristic subgroup.
Proof
Handson proof
Given: A group , such that acts transitively on the set of nonidentity elements of . A characteristic subgroup of
To prove: Either is trivial, or
Proof: If is trivial, we are done. Otherwise, there exists a nonidentity 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.
Conceptual 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.