Every finite group occurs as the automorphism group of at most finitely many finite groups
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.
- It is possible that there is no finite group whose automorphism group is .
- There may exist infinite groups whose automorphism group is . For instance, the automorphism group of the group of integers is the cyclic group of order two.
- Trivial automorphism group implies trivial or order two
- More facts in Category:Results about control via the automorphism group
Similar and opposite facts about inner and outer automorphism groups
- Every finite cyclic group occurs as the outer automorphism group of a finite simple non-abelian group (and in fact, of infinitely many finite simple non-abelian groups, since the construction has an arbitrary prime)
- Every countable group occurs as the outer automorphism group of a finitely generated group