# Centerless implies NSCFN in automorphism group

From Groupprops

## Statement

Suppose is a Centerless group (?). Then, consider the homomorphism , given by the action by conjugation. Then, the homomorphism is injective. Identifying with its image, , we obtain that is a subgroup of . This subgroup is a NSCFN-subgroup (?), i.e.:

- is a normal subgroup inside .
- is a self-centralizing subgroup inside , i.e., (in fact, is a centralizer-free subgroup).
- is a fully normalized subgroup of : Every automorphism of extends to an inner automorphism of .

## Related facts

- Every group is normal fully normalized in its holomorph
- Abelian implies self-centralizing in holomorph
- Additive group of a field implies monolith in holomorph
- Characteristically simple and non-abelian implies monolith in automorphism group
- Characteristically simple and non-abelian implies automorphism group is complete