Schreier's lemma


In mathematics, Schreier's lemma is a theorem in group theory used in the Schreier–Sims algorithm and also for finding a presentation of a subgroup.

Statement

Suppose is a subgroup of, which is finitely generated with generating set , that is, G =.
Let be a right transversal of in. In other words, is a section of the quotient map, where denotes the set of right cosets of in.
We make the definition that given ∈, is the chosen representative in the transversal of the coset, that is,
Then is generated by the set

Example

Let us establish the evident fact that the group Z3 = Z/3Z is indeed cyclic. Via Cayley's theorem, Z3 is a subgroup of the symmetric group S3. Now,
where is the identity permutation. Note S3 =.
Z3 has just two cosets, Z3 and S3 \ Z3, so we select the transversal, and we have
Finally,
Thus, by Schreier's subgroup lemma, generates Z3, but having the identity in the generating set is redundant, so we can remove it to obtain another generating set for Z3, .