What examples are there of a group in which every element is nilpotent and has infinitely many square roots?

Question

What examples are there of a group in which every element:

• is nilpotent
• has infinitely many square roots

(full disclosure: I recently asked a similar question but made a mistake of failing to specify nilpotency of the elements.)

By square roots, I mean $a$ is a square root of $b$ if $a\cdot a=b$ (apologies if that is obvious).

By nilpotent, I mean for every element $a$ there exists integer $n$ such that $a^n=e$ where $e$ is the group identity.

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What have I tried? Well I'm a novice at group theory so this is tricky for me. But I think this much:

• Prufer p-group has all nilpotent elements, but every element in these has only $p$ many $p^{th}$ roots and I need infinitely many. So it would seem I might be looking for a direct product or sum of up to infinitely many Prufer p-groups.
• Since every element is a square root and is nilpotent, it follows every element $a$ must satisfy $a^{2^n}=e$
• This makes me think I need a product of infinitely many Prufer 2-groups

Answer 1

The condition you're calling nilpotence is called being torsion or having finite order; nilpotence in group theory refers to a condition on a group, not on an element of a group.

So, you're looking for a torsion group in which every element has infinitely many square roots. If the group is also abelian, then this condition is equivalent to requiring that every element has at least one square root and that there are infinitely many elements of order $2$. So as Hagen von Eitzen says in the comments, the direct sum of infinitely many copies of the Prufer group $\mathbb{Z}[1/2]/\mathbb{Z}$ will work.