Consider a group presentation given by n generators xi and a single cyclically reduced relator r. If x1 appears in r, then (according to the freiheitssatz) 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.
The result was proposed by the German mathematician Max Dehn and proved by his student, Wilhelm Magnus, in his doctoral thesis.
[1] Although Dehn expected Magnus to find a topological proof,[2] Magnus instead found a proof based on mathematical induction[3] and amalgamated products of groups.
[3][5][6] The freiheitssatz has become "the cornerstone of one-relator group theory", and motivated the development of the theory of amalgamated products.