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Answers
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Example of Noetherian group (every subgroup is finitely generated) that is not finitely presented
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What part of the fundamental group is captured by the second homology group?
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How about the Lie algebra over commutative ring?
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Does any research mathematics involve solving functional equations?
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Exponent of a group
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Group cannot be the union of conjugates
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References for functional equations in more general settings than the reals