The Tall Tale of Terminating Transfinite Towers

Question

The transfinite tower of iterative automorphisms of a group $G$ is simply definied to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the direct limit is taken at the limit stages:

$G\rightarrow Aut(G)\rightarrow Aut(Aut(G))\rightarrow\cdots\rightarrow G_{\alpha}\rightarrow G_{\alpha+1}\rightarrow\cdots$

The tower terminates when a fixed point is reached, namely one of the groups in the chain is isomorphic to its automorphism group by the natural map. Simon Thomas has proved that the automorphism tower of every centerless group eventually terminates. Later, Hamkins completed Thomas' result by showing that the automorphism tower terminates for every group:

• Thomas, Simon, The automorphism tower problem, Proc. Am. Math. Soc. 95, 166-168 (1985). ZBL0575.20030.

• Hamkins, Joel David, Every group has a terminating transfinite automorphism tower, Proc. Am. Math. Soc. 126, No. 11, 3223-3226 (1998). ZBL0904.20027.

Hamkins' theorem gives a sense to the natural definition of the notion of terminating number of a group, $\tau(G)$, that is the least ordinal where the automorphism tower of $G$ terminates.

My first question is about the minimum power of $ZFC$ that is needed to carry out Thomas-Hamkins' proof:

Question 1. How much $ZFC$ is needed to prove that the automorphism tower terminates for every group, $G$, and so $\tau(G)$ is well-defined? Particularly, is $AC$ used anywhere in Hamkins or Thomas' results (which Hamkins' proof is partially based on)? If so, is this use of $AC$ essential? If yes, are the following two statements equivalent?

• The automorphism tower terminates for every group.

• The Axiom of Choice.

My next question is about the relation between the terminating number of the direct product of two groups and the terminating number of each component:

Question 2. What is the relation between $\tau (G\times H)$ and $\tau (G)$, $\tau(H)$? Is there an upper bound for $\tau (G\times H)$ expressible in terms of $\tau (G)$, $\tau(H)$? For instance, is it true to say $\tau (G\times H)\leq Max (\tau (G), \tau(H))$ or $\tau (G)+\tau(H)$ or $\tau (G).\tau(H)$ ...?

The "Max" bound in the above question is inspired by the fact that for finite groups, $G, H$, whose orders are relatively prime, we have $Aut(G\times H)\cong Aut(G)\times Aut(H)$. If one somehow manages to keep this pattern through the entire chain then the automorphism tower of $G\times H$ terminates after $Max (\tau (G), \tau(H))$ steps.

In particular, computing $\tau(G^n)$ (and comparing it with $\tau(G)$) could be of interest as well. For instance, in the special case that $G$ is a cyclic group of order $p$, one has $Aut(G^n)\cong GL_{n}(\mathbb{F}_p)$ and so $\tau (G^{n})=\tau (GL_{n}(\mathbb{F}_p))+1$.

Answer 1

Here is a relevant paper,

Kaplan, Itay; Shelah, Saharon, The automorphism tower of a centerless group without choice, Arch. Math. Logic 48, No. 8, 799-815 (2009). ZBL1192.03026.

At least for a centerless group, it seems that every automorphism tower stabilizes below the Lindenbaum number of the power set of finite sequences of the group.

While I don't quite know how to move from a group to a centerless group without choice, let me point out that $G\mapsto\operatorname{Aut}(G)$ is a function on the universe, and by replacement there is some $\alpha$ such that $V_\alpha$ is a closure point of that function. I suspect that an argument would work in a choiceless setting from this point.