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Then, the inverse image in <math>\operatorname{Aut}(G)</math> of the unique minimal subgroup of <math>\operatorname{Out}(G)</math> 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 <math>r</math> is the outer automorphism group of the [[projective special linear group of degree two]] <math>PSL(2,2^r)</math>. In particular, when <math>r = p^k</math> for some [[prime number]] <math>p</math> and <math>k \ge 1</math>, we obtain situations where the theorem applies.
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