# 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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## History

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.

## Statement

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.