Freiheitssatz
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In mathematics, the Freiheitssatz (German: "freedom/independence theorem") 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.
[edit] Statement of the theorem
Consider a group presentation
given by n generators xi and a single cyclically reduced relator r. If x1 appears in r, then the subgroup of G generated by x2, ..., xn is a free group, freely generated by x2, ..., xn. In other words, the only relations involving x2, ..., xn are the trivial ones.
[edit] External links
- V.A. Roman'kov (2001), “Freiheitssatz”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
[edit] References
- Magnus, Wilhelm (1930). "Über discontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz)". J. Reine Angew. Math. 163: 141–165.
