Finite groups in which every element is a commutator are considered in many works. How about infinite group case? Are there any recent results or constructions related to infinite groups in which every element is a commutator?
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1$\begingroup$ Take a product of infinitely many non-trivial finite groups in which every element is a commutator. What is the question here? $\endgroup$– zntCommented Dec 5, 2016 at 9:49
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$\begingroup$ This becomes interesting when you assume finite generation of the group. $\endgroup$– MishaCommented Dec 5, 2016 at 9:58
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4$\begingroup$ There are infinite groups in which all non-identity elements are conjugate. It is immediate that every element is a commutator in such a group. $\endgroup$– Geoff RobinsonCommented Dec 5, 2016 at 11:42
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1$\begingroup$ @GeoffRobinson And even finitely presented ones. arxiv.org/pdf/math/0411039v3.pdf $\endgroup$– Denis TCommented Dec 5, 2016 at 15:19
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1$\begingroup$ @ArturoMagidin I think you are missing something in your comment, and it is not obvious to me why Tarski monsters have this property (It is clear the commutator subgroup is the whole group but it isn't clear that every element is a commutator to me) $\endgroup$– user35370Commented Dec 5, 2016 at 19:13
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