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I understand that the only free groups that are abelian are 1 and Z, hence a difference between 'free abelian groups' and 'abelian free groups'.
Can someone please tell me what are the solvable free groups?
How can I construct 'free solvable groups'?
What is known about free groups in other classes of groups like polycyclic, nilpotent ...?

Thanks.

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    Are you aware of the abstract definition of free object which is implicitly being used in both of the answers below?2010-10-16

2 Answers 2