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

## 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.

## 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.