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I was just working on a problem for my Algebra class about proving that a group was free. I was able to figure that out, but it got me wondering: are there any properties that are unique to free groups that are useful in proving that a group is free? Or a few properties that are only all satisfied in a free group?

That may have been a little vague, and I am sorry about that. I know that this kind of vagueness is not desirable on MSE, but I am really just curious about this :)

EDIT: I am interested in finitely generated free groups

Anonymous
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  • Which definition of a free group are you using? – Shaun Apr 13 '20 at 20:59
  • @Shaun I am talking about non-abelian free groups – Anonymous Apr 13 '20 at 21:12
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    Still, free groups have more than one definition. They are, of course, equivalent, but which one you use would determine what unique properties - properties in alternative definitions - they have. – Shaun Apr 13 '20 at 21:14
  • @Shaun I am considering $F_n = \langle a_1, a_2, \dots, a_n \rangle$ where there are no relations between the generators. – Anonymous Apr 13 '20 at 21:16
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    So, do they have to be finitely generated? – Shaun Apr 13 '20 at 21:19
  • @Shaun Yes, I am interested in finitely generated free groups. I will update the question! – Anonymous Apr 13 '20 at 21:20
  • [Related.](https://math.stackexchange.com/q/855321/104041) – Shaun Apr 13 '20 at 21:23
  • @Shaun this is probably completely wrong, but I heard once that Hopfian, imperfect, and finitely generated implies free. Is this true? – Anonymous Apr 13 '20 at 21:27
  • The closest thing I can think of in that regard is that every finitely generated free group is Hopfian. This is **Proposition 3.5** of Lyndon & Schupp. – Shaun Apr 13 '20 at 21:30
  • @Shaun not super related, but is it known exactly which families of groups are Hopfian? – Anonymous Apr 13 '20 at 21:36
  • I don't know (yet), @Anonymous. I'll look into it. – Shaun Apr 13 '20 at 21:40
  • "Hopfian, imperfect, and finitely generated implies free" is very much false. For instance, finite rank abelian groups, fundamental groups of surfaces of genus $\ge 1$, etc, all share these properties. – Moishe Kohan Apr 14 '20 at 03:36

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