Nielsen–Schreier theorem

The set of Nielsen - Schreier is a fundamental result of combinatorial group theory, a branch of mathematics that deals with discrete (usually infinite) groups. The theorem states that in a free group every subgroup is free. In addition to this qualitative statement, the quantitative version of a relationship here between the index and the rank of a sub-group. This has the surprising consequence that a free group of rank subgroups of any rank, and even of ( countable ) infinite rank has.

The set may algebraic- topological methods proved particularly elegant and vividly with the help, by the fundamental group and superpositions of graphs.

• 3.1 subgroups of integers
• 3.2 subgroups non- abelian free groups
• 3.3 Subgroups of finitely generated groups

Statement of the theorem

Is a free group, then every subgroup of is also free. This qualitative statement is true in general; in the finite case is also considered the following quantitative statement: Is a free group of finite rank and is a subgroup of finite index, then is free of rank.

Evidence

The theorem can be proved either algebraic or topological arguments. The topological proof is considered particularly elegant, and will be outlined below. It uses a sophisticated way the representation of free groups as fundamental groups of graphs and is a prime example of fruitful interaction between algebra and topology.

Free groups as fundamental groups of graphs

Be a connected graph. We realize this as a topological space, where each edge corresponds to a path between the adjacent corners. The crucial observation now is that the fundamental group is a free group.

To make this result explicit, and thus to prove choose a maximal tree, ie a tree that contains all the corners of. The remaining edges provide a basis of, by choosing a path for each edge that runs from the base point in the tree to the edge, crossed it, and then returns back to the base point. ( We choose as base point expediently a corner, . These is then automatically in each maximal tree ) The fact that form the homotopy classes with a base of, one can by means of combinatorial homotopy prove or by explicit construction of the universal covering of the graph.

This result we can quantitatively summarize, if a finite graph with vertices and edges. He then has the Euler characteristic. Each maximum tree then contains exactly corners and edges, and in particular the Euler characteristic. There remain the edges and their number is. The fundamental group is therefore a free group of rank.

Topological proof of the theorem of Nielsen - Schreier

Qualitative version: Every free group can be represented as the fundamental group of a graph. For each sub- group is an overlay. The overlay area is again a graph, so the group is free.

Quantitative version: Every free group of finite rank can be represented as the fundamental group of a finite graph with Euler characteristic. For each subgroup of index then heard a - sheeted covering. So the overlapping graph has the Euler characteristic, and the group is therefore free of rank.

Conclusions

Among groups of integers

For the rank is the trivial group consisting only of the identity element, and the statement of the theorem is empty.

The first interesting application, we find the rank. Here is the free abelian group, and we find the classification of subgroups of re: The trivial subgroup is free of rank, any other subgroup is of the form of the index and self- released abelian of rank.

Subgroups of non- abelian free groups

For a free group of rank follows from the (quantitative) set of Nielsen - Schreier that free subgroups of arbitrary finite rank contains. (One can even construct a sub-group of countably infinite rank. )

This amazing property is in contrast to free abelian groups (where the rank of a sub-group is always smaller or equal to the rank of the entire group), or vector spaces over a field (where the dimension of a subspace is always less than or equal to the dimension of the entire room is ).

Subgroups of finitely generated groups

Although the set of Nielsen - Schreier is initially only of free groups, but its quantitative version also has interesting consequences for arbitrary finite groups generated. If a group is finitely generated ( by a family with elements of ), and is a subgroup of finite index, then is finitely generated ( by a family with elements of ).

As in the case of free groups so you have to expect that a subgroup generator requires more than the whole group in general.

History

The set is named after the mathematicians Jakob Nielsen and Otto Schreier. It was proved in 1921 by Nielsen, initially only for free groups of finite rank. Schreier could remove this restriction in 1927, and generalize the theorem to arbitrary groups. Max Dehn recognized relations with algebraic topology and gave the first a topological proof of the theorem of Nielsen - Schreier. Kurt Reidemeister put this development in 1932 in his textbook on combinatorial topology dar.