Freiheitssatz
Encyclopedia
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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Freiheitssatz (German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
: "freedom/independence theorem": Freiheit + Satz
Satz (disambiguation)
Satz is a German word and name, and may refer to:* Satz, a formal section in music analysis* "theorem" in German, conventionally used in the name of certain theorems. Note however that the word "Theorem" is also used in German, generally for more important results, and thus in a stricter...
) is a result in the presentation theory
Presentation of a group
In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...
of groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
. The result was proposed by the German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Max Dehn
Max Dehn
Max Dehn was a German American mathematician and a student of David Hilbert. He is most famous for his work in geometry, topology and geometric group theory...
and proved by his student, Wilhelm Magnus
Wilhelm Magnus
Wilhelm Magnus was a mathematician. He made important contributions in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations....
, in his doctoral thesis.
Statement of the theorem
Consider a group presentationgiven by n generators xi and a single cyclically reduced relator r. If x1 appears in r, then the subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of G generated by x2, ..., xn is a free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
, freely generated by x2, ..., xn. In other words, the only relations involving x2, ..., xn are the trivial ones.