This was inspired by a discussion with James Aaronson outside the Chapel. We were contemplating group theory and automorphisms (specifically, trying to determine whether there exists a group of size strictly larger than with only two automorphisms), and eventually considered the following question:

What happens if you begin with a group G, and iteratively apply the ‘automorphism group’ operator?

For a cyclic group of order *n*, where *n* has primitive roots, the automorphism group is the cyclic group of order φ(*n*). If a group is non-Abelian, its automorphism group is also non-Abelian (consider automorphisms arising from conjugation by elements which do not commute). Similarly, if a group is Abelian and non-cyclic, its automorphism group is non-Abelian. Hence, the groups that eventually reduce to the trivial group under repeated application of Aut are cyclic groups of order *n* such that {*n*, φ(*n*), φ(φ(*n*)), …, 1} all have primitive roots. James Aaronson completely characterised the integers with this property:

- , for all ;
- ;
- Primes
*p* of the form ;
- 2
*p*, where *p* is a prime of that form;
- 4, 5, 10, 11, 22, 23, 46, 47 and 94.

These form the sequence A117729.

The next case is C8, which has no primitive roots. Its automorphism group is the Klein 4-group, C2 × C2. Since it is the additive group of a vector space over F2, the automorphism group is the group of linear maps (2 × 2 invertible matrices over F2 with linearly independent columns). It is not too hard to show that this is isomorphic to S3 (symmetric group with 6 elements), and that the direct sum has the general linear group GL(*n, p*) as its automorphism group.

Most symmetric groups (the exceptions being *n* = 2 and *n* = 6) are their own automorphism groups. The next few cases, such as D8 and Q8, are included in the diagram below:

As the size of the group increases, these become difficult to compute.

For simple non-Abelian groups, the automorphism group is complete (equal to its own automorphism group). This is proved here. More generally, for centreless groups, it is possible to define a limit of a sequence of iterated automorphism groups, known as an *automorphism tower*. It has been proved that this process eventually terminates at some ordinal (not necessarily finite).

This is discussed in further detail on Math Overflow.

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Actually, the automorphism group of C15 is C2*C4, which is not cyclic, so the statement made in the third paragraph is incorrect

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Lots of related stuff, including a proof of the difficult Wielandt’s theorem, can be found here: http://www.math.rutgers.edu/~sthomas/book.ps