Is every subgroup of a finitely generated group also finitely generated? And if so does it have the same amount of generators?
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No, see http://math.stackexchange.com/questions/983480/commutator-subgroup-of-rank-2-free-group-is-not-finitely-generated – Alessandro Codenotti Nov 10 '16 at 19:16
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absolutely amazing, thank you – Nov 10 '16 at 19:17
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Regarding the second question even if the subgroup is finitely generated it can have fewer generators than the whole group, just take the cyclic group generated by one of them! – Alessandro Codenotti Nov 10 '16 at 19:37