Given a set $X$ is it provable in $\mathsf{ZF}$ that there is a binary operation $\ast: X\times X\to X$ such that $(X,\ast)$ is a group?
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No, as this is equivcalent to AC. See http://mathoverflow.net/questions/12973/does-every-non-empty-set-admit-a-group-structure-in-zf This is thus a duplicate. – Nov 28 '14 at 12:11
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Oh - it is a duplicate. I'm sorry. – Dominic van der Zypen Nov 28 '14 at 12:11
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And in any case, you need to assume $X$ is nonempty! – Arturo Magidin Nov 28 '14 at 22:30