Every automorphism is center-fixing and outer automorphism group is rank one p-group implies not every normal-extensible automorphism is inner

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Definition

Suppose G is a group satisfying both the following conditions:

  1. \operatorname{Out}(G), 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).
  2. Every automorphism of G is a center-fixing automorphism.

Then, the inverse image in \operatorname{Aut}(G) of the unique minimal subgroup of \operatorname{Out}(G) is contained in the group of normal-extensible automorphisms. In particular, not every normal-extensible automorphism is inner.

Related facts

Examples

Every finite cyclic group occurs as the outer automorphism group of a finite simple non-abelian group. More specifically, the cyclic group of order r is the outer automorphism group of the projective special linear group of degree two PSL(2,2^r). In particular, when r = p^k for some prime number p and k \ge 1, we obtain situations where the theorem applies.