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.

- Does there exist a periodic group (duplicate of Periodic Automorphism Towers )?
- Does there exist a divergent group (
**not**a duplicate of Does $\mathrm{Aut}(\mathrm{Aut}(...\mathrm{Aut}(G)...))$ stabilize? because the answer does not consider the group merely up to isomorphism)? - What is the eventual behaviour of the prime cyclic group $C_{41}$ (the smallest prime cyclic group whose behaviour I have been unable to track)?

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?