Nielsen–Schreier theorem

In group theory, a branch of mathematics, the Nielsen–Schreier theorem states that every subgroup of a free group is itself free.^[1][2][3] It is named after Jakob Nielsen and Otto Schreier.

Statement of the theorem

A free group may be defined from a group presentation consisting of a set of generators and the empty set of relations (equations that the generators satisfy). That is, it is the unique group in which every element is a product of some sequence of generators and their inverses, and in which there are no equations between group elements that do not follow in a trivial way from the equations gg⁻¹ describing the relation between a generator and its inverse. The elements of a free group may be described as all of the possible reduced words; these are strings of generators and their inverses, in which no generator is adjacent to its own inverse. Two reduced words may be multiplied by concatenating them and then removing any generator-inverse pairs that result from the concatenation.

The Nielsen–Schreier theorem states that if G is a subgroup of a free group, then G is itself isomorphic to a free group. That is, there exists a subset S of elements of G such that every element in G is a product of members of S and their inverses, and such that S satisfies no nontrivial relations.

The Nielsen–Schreier formula, or Schreier index formula, quantifies the result in the case where the subgroup has finite index: if G is a free group on n generators, and H is a subgroup of finite index e, then H is free of rank^[4]

Example

Let G be the free group with two generators, a and b, and let E be the subgroup consisting of all reduced words that are products of evenly many generators or their inverses. Then E is itself generated by the six elements p = aa, q = ab, r = ab⁻¹, s = ba, t = ba⁻¹, and u = bb. A factorization of any reduced word in E into these generators and their inverses may be constructed simply by taking consecutive pairs of symbols in the reduced word. However, this is not a free presentation of E because it satisfies the relations p = qr⁻¹ = rq⁻¹ and s = tu⁻¹ = ut⁻¹. Instead, E is generated as a free group by the three elements p = aa, q = ab, and s = ba. Any factorization of a word into a product of generators from the six-element generating set {p, q, r, s, t, u} can be transformed into a product of generators from this smaller set by replacing r with ps⁻¹, replacing t with sp⁻¹, and replacing u with sp⁻¹q. There are no additional relations satisfied by these three generators, so E is the free group generated by p, q, and s.^[5] The Nielsen–Schreier theorem states that this example is not a coincidence: like E, every subgroup of a free group can be generated as a free group, possibly with a larger set of generators.

Proof

It is possible to prove the Nielsen–Scheier theorem using topology.^[1] A free group G on a set of generators is the fundamental group of a bouquet of circles, a topological graph with a single vertex and with an edge for each generator.^[6] Any subgroup H of the fundamental group is itself a fundamental group of a covering space of the bouquet, a (possibly infinite) topological Schreier coset graph that has one vertex for each coset of the subgroup.^[7] And in any topological graph, it is possible to shrink the edges of a spanning tree of the graph, producing a bouquet of circles that has the same fundamental group H. Since H is the fundamental group of a bouquet of circles, it is itself free.^[6] This proof is due to Reinhold Baer and Friedrich Levi (1936); the original proof by Schreier forms the Schreier graph in a different way as a quotient of the Cayley graph of G modulo the action of H.

According to Schreier's subgroup lemma, a set of generators for a free presentation of H may be constructed from cycles in the covering graph formed by concatenating a spanning tree path from a base point (the coset of the identity) to one of the cosets, a single non-tree edge, and an inverse spanning tree path from the other endpoint of the edge back to the base point.^[8]

Axiomatic foundations

Although several different proofs of the Nielsen–Schreier theorem are known, they all depend on the axiom of choice. In the proof based on fundamental groups of bouquets, for instance, the axiom of choice appears in the guise of the statement that every connected graph has a spanning tree. The use of this axiom is necessary, as there exist models of Zermelo–Fraenkel set theory in which the axiom of choice and the Nielsen–Schreier theorem are both false. The Nielsen–Schreier theorem in turn implies a weaker version of the axiom of choice, for finite sets.^[9]

History

The Nielsen–Schreier theorem is a non-abelian analogue of an older result of Richard Dedekind, that every subgroup of a free abelian group is free abelian.^[3]

Jakob Nielsen (1921) originally proved a restricted form of the theorem, stating that any finitely-generated subgroup of a free group is free. His proof involves performing a sequence of Nielsen transformations on the subgroup's generating set that reduce their length (as reduced words in the free group from which they are drawn).^[1][10] Otto Schreier proved the Nielsen–Schreier theorem in its full generality in his 1926 habilitation thesis, Die Untergruppen der freien Gruppe, also published in 1927 in Abh. math. Sem. Hamburg. Univ.^[11][12]

The topological proof based on fundamental groups of bouquets of circles is due to Reinhold Baer and Friedrich Levi (1936). Another topological proof, based on the Bass–Serre theory of group actions on trees, was published by Jean-Pierre Serre (1970).^[13]

See also

• Fundamental theorem of cyclic groups, a similar result for cyclic groups that in the infinite case may be seen as a special case of the Nielsen–Schreier theorem

Notes

References

• Baer, Reinhold; Levi, Friedrich (1936), "Freie Produkte und ihre Untergruppen", Compositio Mathematica 3: 391–398 .
• Fried, Michael D.; Jarden, Moshe (2008), Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 11 (3rd ed.), Springer-Verlag, p. 70, ISBN 978-3-540-77269-9, Zbl 1145.12001 .
• Howard, Paul E. (1985), "Subgroups of a free group and the axiom of choice", The Journal of Symbolic Logic 50 (2): 458–467, doi:10.2307/2274234, MR 793126 .
• Johnson, D. L. (1980), Topics in the Theory of Group Presentations, London Mathematical Society lecture note series 42, Cambridge University Press, ISBN 978-0-521-23108-4 .
• Johnson, D. L. (1997), Presentations of Groups, London Mathematical Society student texts 15 (2nd ed.), Cambridge University Press, ISBN 978-0-521-58542-2 .
• Läuchli, Hans (1962), "Auswahlaxiom in der Algebra", Commentarii Mathematici Helvetici 37: 1–18, doi:10.1007/bf02566957, MR 0143705 .
• Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald (1976), Combinatorial Group Theory (2nd revised ed.), Dover Publications .
• Nielsen, Jakob (1921), "Om regning med ikke-kommutative faktorer og dens anvendelse i gruppeteorien", Math. Tidsskrift B (in Danish) 1921: 78–94, JFM 48.0123.03 .
• Rotman, Joseph J. (1995), An Introduction to the Theory of Groups, Graduate Texts in Mathematics 148 (4th ed.), Springer-Verlag, ISBN 978-0-387-94285-8 .
• Serre, J.-P. (1970), Groupes Discretes, Extrait de I'Annuaire du College de France, Paris .
• Stillwell, John (1993), Classical Topology and Combinatorial Group Theory, Graduate Texts in Mathematics 72 (2nd ed.), Springer-Verlag .