Iterated automorphism groups of finite groups

Let $\mathcal{G}$ be the set of isomorphism classes of finite groups.

There is an operation $\mathrm{Aut} : \mathcal{G} \rightarrow \mathcal{G}$ which gives the automorphism group of a given group, up to isomorphism. In the most general setting, my question is:

What possible eventual behaviour can arise from iterating the operation $\mathrm{Aut}$?

More precisely, there is a trichotomy of possible eventual behaviours:

• Static: a group such as $D_8$ or $S_8$, which is isomorphic to its automorphism group.
• Periodic: a group which enters a cycle of length $\geq 2$.
• Divergent: a group which never enters a cycle, and therefore grows without bound.

I can exhibit lots of examples of static groups, and groups which eventually evolve into static groups after many steps, such as:

$$C_{2879} \mapsto C_{2878} \mapsto C_{1438} \mapsto C_{718} \mapsto C_{358} \mapsto C_{178} \mapsto C_{88} \\ \mapsto C_2 \times C_2 \times C_2 \times C_5 \mapsto PSL(2,7) \times C_4 \mapsto PGL(2, 7) \times C_2$$

However, I don't have any examples of periodic or divergent groups.

I conjecture that $C_{41}$ does diverge, simply because after several iterations it spawns a direct sum involving lots of copies of $C_2$:

$$C_{41} \mapsto C_{40} \mapsto C_4 \times C_2 \times C_2 \mapsto D_4 \mapsto F_4/Z \times C_2 \\ \mapsto (S_4 \wr C_2) \times C_2 \\ \mapsto (S_4 \wr C_2) \times C_2 \times C_2 \\ \mapsto (S_4 \wr C_2) \times S_4 \times C_2 \times C_2 \\ \mapsto (S_4 \wr C_2) \times S_4 \times S_4 \times C_2 \times C_2 \times C_2 \times C_2$$

Finally, for the most ambitious question of all:

• Does there exist an algorithm which, when given a description of a Turing machine, outputs a finite group which is divergent if and only if the Turing machine does not halt?
• For your last question, in what form do you want the finite group to be given? If only as a presentation, then one can do this just by considering that the question of whether a given presentation is the trivial group or not is Turing equivalent to the halting problem. – Joel David Hamkins Mar 2 '18 at 13:09
• Another relevant question: math.stackexchange.com/questions/1096939/… – Nick Gill Mar 2 '18 at 14:03
• @JoelDavidHamkins It's finite, so it may as well be given as a Cayley table. – Adam P. Goucher Mar 2 '18 at 18:03
• The group, $Aut^3 (C_{41})$ is not $D_4$, but a group of order 192 (with GAP Small Groups Library id of 1493 for groups of that order). I got as far as $Aut^8$, but could not verify stabilization by that step (it may be worth test if the group or any of its predecessors are centerless as then Wielandt's classic 1939 theorem would give stabilization in finitely many steps). All examples I have worked on (every group of order<64) has either stabilized or blown up beyond what I can compute in GAP. So periodic behavior would be the most surprising to see. – Justin Benfield Nov 30 '19 at 12:51
• Decided to do the center check myself, $Aut^8(G)$ has center $C_2$, so the question remains open for $C_{41}$. – Justin Benfield Nov 30 '19 at 13:21