Every finite group occurs as the automorphism group of at most finitely many finite groups

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This article gives a result about how information about the structure of the automorphism group of a group (abstractly, or in action) can control the structure of the group
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The result was proved by Hariharan K. Iyer in his paper On Solving the Equation Aut(X) = G in the Rocky Mountain Journal of Mathematics, Volume 9, No. 4, Fall 1979.


Let G be a finite group. Then, there are at most finitely many finite groups H such that \operatorname{Aut}(H) = G.

Note that:

  • It is possible that there is no finite group whose automorphism group is G.
  • There may exist infinite groups whose automorphism group is G. For instance, the automorphism group of the group of integers is the cyclic group of order two.

Related facts

Similar and opposite facts about inner and outer automorphism groups


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