Computable Group Copies without Computable Isomorphism

Question

What's an explicit example of isomorphic and computable groups without computable isomorphisms? Just from reading through theorems on the internet I know they must exist, but I can't quite think of an example myself.

Answer 1

First, we show the following:

Let $G$ be the usual copy of the group $\mathbb{Z}$. We can find, computably in $e$, a computable copy $H_e$ of $\mathbb{Z}$ such that $\Phi_e$ is not an isomorphism from $G$ to $H$.

This is a direct "wait-and-break": $H_e$ will look like $G$ until we see $\Phi_e(1)\downarrow=k$ for some $k$ (if this never happens then $G=H$ but $\Phi_e$ is not total, so that's not a problem). When we do, if $k\not=1$ we just keep building $H=G$; if $k=1$ we treat what we've built as part of $2\mathbb{Z}$ as opposed to $\mathbb{Z}$ and "stick new points in" accordingly.

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Next, we view the above observation as the $e$th "atomic module" and combine the constructions above to satisfy all of our atomic modules at once:

Let $G$ be the usual copy of the group $\bigoplus_{e\in\mathbb{N}}\mathbb{Z}.$ There is a computable copy $H$ of $G$ which is not computably isomorphic to $G$.

This is more subtle than it may look at first glance - keep in mind that a putative isomorphism need not "respect axes," so we can't just literally "copy-and-paste" the observation above infinitely many times. But it's ultimately the same idea, just messier. (And we do need to "go up to infinite dimensions" - for each finite $n$ the group $\mathbb{Z}^n$ is computably categorical, as is $\mathbb{Q}$.)