Freiheitssatz

In mathematics, the Freiheitssatz (German: "freedom/independence theorem": Freiheit + Satz) is a result in the presentation theory of groups. The result was proposed by the German mathematician Max Dehn and proved by his student, Wilhelm Magnus, in his doctoral thesis.

Statement of the theorem

Consider a group presentation

$G = \langle x_{1}, \dots, x_{n} | r = 1 \rangle$

given by n generators x_i and a single cyclically reduced relator r. If x₁ appears in r, then the subgroup of G generated by x₂, ..., x_n is a free group, freely generated by x₂, ..., x_n. In other words, the only relations involving x₂, ..., x_n are the trivial ones.

References

• Magnus, Wilhelm (1930). "Über diskontinuierliche Gruppen mit einer definierenden Relation. (Der Freiheitssatz)". J. Reine Angew. Math. 163: 141–165.

External links

• V.A. Roman'kov (2001), "Freiheitssatz", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4