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Is there a name for groups $G$ which satisfy the property that for any $a$ and $b$ in $G$, there is a $c\in G$ and integers $n$ and $m$ such that $a=c^n$ and $b=c^m$? Such a group has to be abelian, and it's not hard to prove that this is equivalent to the property that every finitely generated subgroup is cyclic.

As a generalization, has anyone studied groups $G$ in which for any three elements $a,b,c\in G$, there are two other elements $x$ and $y$ such that $a$, $b$ and $c$ all lie in the subgroup generated by $x$ and $y$?

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    $\begingroup$ en.wikipedia.org/wiki/Locally_cyclic_group $\endgroup$ – Mustafa Gokhan Benli Oct 3 '13 at 20:40
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    $\begingroup$ An example for the groups you described in your first paragraph is $\mathbb{Q}$. Since these groups are abelian, I suspect module theory over $\mathbb{Z}$ would be a better framework to study them. $\endgroup$ – user23860 Oct 3 '13 at 20:43
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    $\begingroup$ These are precisely subquotients of $\mathbb{Q}$. Also they can be described as subgroups of $\mathbb{Q}$ and $\mathbb{Q}/\mathbb{Z}$. $\endgroup$ – YCor Oct 3 '13 at 21:07
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    $\begingroup$ Groups in which "every finitely generated subgroup is $\mathfrak{X}$" are called "locally $\mathfrak{X}$ groups". $\endgroup$ – Arturo Magidin Oct 3 '13 at 22:03
  • $\begingroup$ Your generalized class of groups consists of all direct limits of 2-generator groups (including all 2-generator groups). Since every countable group embeds in a 2-generator group, I suspect that every group embeds in a group in your class. (There are some set-theoretic issues to worry about, but I don't see an obstruction.) So it's hard to imagine what theorems can be proved about your class. $\endgroup$ – HJRW Oct 6 '13 at 8:28

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