Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible
- Every automorphism of is a center-fixing automorphism: it fixes every element of the center .
- is a maximal subgroup of .
- Centerless and maximal in automorphism group implies every automorphism is normal-extensible
- Finite p-group with center of prime order and inner automorphism group maximal in p-Sylow-closure of automorphism group implies every p-automorphism is p-normal-extensible
- Every automorphism is center-fixing and outer automorphism group is rank one p-group implies not every normal-extensible automorphism is inner
Given: A group such that every automorphism of fixes every element of , and is a maximal subgroup of .
To prove: For any automorphism of , and any group containing as a normal subgroup, extends to an automorphism of .
Proof: Let be the homomorphism given by the action of on by conjugation. This map exists because is normal in .
- The image of is either or : The image of is a subgroup of . It contains , because equals . Since is a maximal subgroup of , the image is either or .
- If the image is , then every automorphism of extends to an inner automorphism of , and we are done. (We say in this case that is a normal fully normalized subgroup of ).
- If the image is , then is a central factor of , i.e., (see fact (2)). Thus, by fact (1), any automorphism that fixes the center of extends to an automorphism of . Since we assumed that every automorphism fixes the center of , we obtain that every automorphism of extends to an automorphism of .