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Reverse Mathematics of the Nielsen-Schreier Theorem

Status: published in *Proceedings of International Conferences on
Mathematical Logic*, Novosibirsk State University Press, 2002, pp. 59 -
71.

Availability: preprint

**Abstract.** The Nielsen-Schreier Theorem states that every
subgroup of a free group is free. To formalize this theorem in weak
subsystems of second order arithmetic, one has to choose between
defining a subgroup in terms of a set of group elements and defining
it in terms of a set of generators. We show that if subgroups are
defined by sets, then the Nielsen-Schreier Theorem is provable in
RCA_{0}, while if subgroups are defined by generators, the
theorem is equivalent to ACA_{0}.

drh@math.uchicago.edu