Freiheitssatz

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.

tatement of the theorem

Consider a group presentation

:$G\; =\; langle\; x\_\{1\},\; dots,\; x\_\{n\}\; |\; r\; =\; 1\; angle$

given by "n" generators "x"_"i" and a single cyclically reduced relator "r". If "x"₁ appears in "r", then the subgroup of "G" generated by "x"₂, ..., "x"_"n" is a free group, freely generated by "x"₂, ..., "x"_"n". In other words, the only relations involving "x"₂, ..., "x"_"n" are the trivial ones.

External links

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References

* cite journal
last = Magnus
first = Wilhelm
authorlink= Wilhelm Magnus
title = Über discontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz)
journal = J. Reine Angew. Math.
volume = 163
year = 1930
pages = 141–165