Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible

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Statement

Suppose G is a group with center Z(G), inner automorphism group \operatorname{Inn}(G), and automorphism group \operatorname{Aut}(G). Suppose the following two statements are true:

Then, every automorphism of G is a normal-extensible automorphism: whenever G is a normal subgroup of some group K and \sigma is an automorphism of G, \sigma extends to an automorphism \sigma' of K.

Related facts

Facts used

  1. Center-fixing implies central factor-extensible: If \sigma is a center-fixing automorphism of a central factor G of a group K, \sigma extends to an automorphism of K.
  2. Equivalence of definitions of central factor

Proof

Given: A group G such that every automorphism of G fixes every element of Z(G), and \operatorname{Inn}(G) is a maximal subgroup of \operatorname{Aut}(G).

To prove: For any automorphism \sigma of G, and any group K containing G as a normal subgroup, \sigma extends to an automorphism of K.

Proof: Let \alpha:K \to \operatorname{Aut}(G) be the homomorphism given by the action of K on G by conjugation. This map exists because G is normal in K.

  1. The image of \alpha is either \operatorname{Inn}(G) or \operatorname{Aut}(G): The image of \alpha is a subgroup of \operatorname{Aut}(G). It contains \operatorname{Inn}(G), because \operatorname{Inn}(G) equals \alpha(G). Since \operatorname{Inn}(G) is a maximal subgroup of \operatorname{Aut}(G), the image is either \operatorname{Inn}(G) or \operatorname{Aut}(G).
  2. If the image is \operatorname{Aut}(G), then every automorphism of G extends to an inner automorphism of K, and we are done. (We say in this case that G is a normal fully normalized subgroup of K).
  3. If the image is \operatorname{Inn}(G), then G is a central factor of K, i.e., K = G C_K(G) (see fact (2)). Thus, by fact (1), any automorphism \sigma that fixes the center of G extends to an automorphism of K. Since we assumed that every automorphism fixes the center of G, we obtain that every automorphism of G extends to an automorphism of K.