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So, I wish to find examples of torsion group where there exist elements of arbitrarily large order. That is a group $G$ where for each $n\in\mathbb{N}$, there exists $g\in G$ such that $g^n = e$ (operation of $G$ applied to $g$ $n$-times), where $e$ is the identity element. I could find one example, whose existence might seem to involve Axiom of Choice. Consider additive groups modulo $n$ and their product over all natural numbers. $$\prod_{n\in\mathbb{N}}\mathbb{Z}_n$$ These are just sequences where $n$-th element is from the additive group $\mathbb{Z}_n$. This is not yet the example, for example $(1,1,1,\ldots,...1,\ldots)$ has infinite order. Consider subgroup $G\leq \prod\mathbb{Z}_n$ of all sequences with finite support. This indeed is a group with elements of arbitrarily large order.

BUT... In order to show there are even elements of $\prod\mathbb{Z}_n$, we must indeed invoke some form of choice. Of course not for elements of $G$. My question is, can some other, possibly simpler, examples be found?

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    $\begingroup$ The most familiar example is $({\mathbb Q}/{\mathbb Z},+)$. $\endgroup$ – Derek Holt Mar 14 at 23:28
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    $\begingroup$ Standard notation to define the subgroup you said is $\bigoplus_{n\in\mathbf{N}}(...)$. By the way, there is no choice at all (I mean, no need to use AC) to show the existence of elements, and in particular in the direct sum! That this direct sum has elements of unbounded order is a theorem of ZF. That you can inject the power set of $\mathbf{N}$ in the whole product is also a theorem of ZF. $\endgroup$ – YCor Mar 15 at 17:21
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Probably the most neat examples of such groups are quasicyclic groups. Not only they do satisfy your condition, but also they are locally cyclic $p$-groups.

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No, no form of choice is needed, as $(0,...,0,...)$ is an element, and $(1,...,1,...)$ as well, and many other examples.

Other examples include, as given in the comments $\mathbb{Q/Z}$, or its "$p$-analogue" $\mathbb Z/p^\infty$, or of course any torsion group containing them (e.g. any direct sum of these)

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