Duality for groups

If F is known to be free, with generators gi, /3 may be constructed by setting j3g; = &», with bi so chosen that pbi = agi. Conversely, let F have the cited property and represent F as a quotient group Fo/Ro> where F0 is a free abelian group. Choose A—F and B = Fo in (1.1), let a be the identity, and p the given homomorphism of .Fo onto F with kernel Ro. Then, by (1.2), a=p/3 is an isomorphism, hence /3 has kernel 0 and thus is an isomorphism of F into F0. Therefore F is isomorphic to a subgroup of a free group F0, so is itself free. The analogous theorem is true for free nonabelian groups, when A and B are interpreted as arbitrary (not necessarily abelian) groups; the proof uses the Schreier theorem [l4] that a subgroup of a free group is free. An abelian group D is said to be infinitely divisible if for each dÇ:D and each integer m there exists in D an element x such that mx = d. Such groups may be characterized by a similar diagram

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