Centerless implies NSCFN in automorphism group

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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).

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