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

From Groupprops

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

View other such results

## Contents

## 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 be a finite group. Then, there are at most finitely many finite groups such that .

Note that:

- 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.

## Related facts

- 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