I am asking for the definition of "free". What is the difference between, say, any old abelian group, and a free abelian group?
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A free abelian group is the same as a product of infinite cyclic group, i.e. a product of copies of $\mathbb{Z}$. A free group is a bit more complicated. – Thibaut Dumont Sep 08 '15 at 14:01
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See also Why are free modules called “free”? – Zev Chonoles Sep 08 '15 at 14:01
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It means there are no constraints on the way generators combine to form new group elements. Such constraints in a non-free group would be given in terms of presentation relations which restrict that some combinations of generators give the identity. In a free group, every combination is unique.
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