Every automorphism is center-fixing and outer automorphism group is rank one p-group implies not every normal-extensible automorphism is inner
Suppose is a group satisfying both the following conditions:
- , the outer automorphism group, is a nontrivial group having a unique minimal subgroup. In other words, it is a p-group of rank one. (when finite, it is either a cyclic group of prime power order or a generalized quaternion group).
- Every automorphism of is a center-fixing automorphism.
Then, the inverse image in of the unique minimal subgroup of is contained in the group of normal-extensible automorphisms. In particular, not every normal-extensible automorphism is inner.
- Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible
- Centerless and maximal in automorphism group implies every automorphism is normal-extensible
Every finite cyclic group occurs as the outer automorphism group of a finite simple non-abelian group. More specifically, the cyclic group of order is the outer automorphism group of the projective special linear group of degree two . In particular, when for some prime number and , we obtain situations where the theorem applies.