# Centerless implies NSCFN in automorphism group

Jump to: navigation, search

## Statement

Suppose $G$ is a Centerless group (?). Then, consider the homomorphism $G \to \operatorname{Aut}(G)$, given by the action by conjugation. Then, the homomorphism is injective. Identifying $G$ with its image, $\operatorname{Inn}(G)$, we obtain that $G$ is a subgroup of $\operatorname{Aut}(G)$. This subgroup is a NSCFN-subgroup (?), i.e.:

1. $G$ is a normal subgroup inside $\operatorname{Aut}(G)$.
2. $G$ is a self-centralizing subgroup inside $\operatorname{Aut}(G)$, i.e., $C_{\operatorname{Aut}(G)}(G) \le G$ (in fact, $G$ is a centralizer-free subgroup).
3. $G$ is a fully normalized subgroup of $\operatorname{Aut}(G)$: Every automorphism of $G$ extends to an inner automorphism of $\operatorname{Aut}(G)$.