# Changes

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

, 22:31, 21 September 2009
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Then, the inverse image in [itex]\operatorname{Aut}(G)[/itex] of the unique minimal subgroup of [itex]\operatorname{Out}(G)[/itex] is contained in the group of [[normal-extensible automorphism]]s. In particular, not every normal-extensible automorphism is inner.

==Related facts==

* [[Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible]]
* [[Centerless and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible]]

==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 [itex]r[/itex] is the outer automorphism group of the [[projective special linear group of degree two]] [itex]PSL(2,2^r)[/itex]. In particular, when [itex]r = p^k[/itex] for some [[prime number]] [itex]p[/itex] and [itex]k \ge 1[/itex], we obtain situations where the theorem applies.