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, . Let be a right transversal of in . In other words, is (the image of) 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 Hence, in particular, Schreier's lemma implies that every subgroup of finite index of a finitely generated group is again finitely generated. 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 = { s1=(1 2), s2 = (1 2 3) }. Z3 has just two cosets, Z3 and S3 \ Z3, so we select the transversal { t1 = e, t2=(1 2) }, and we have Finally, Thus, by Schreier's subgroup lemma, { e, (1 2 3) } generates Z3, but having the identity in the generating set is redundant, so we can remove it to obtain another generating set for Z3, { (1 2 3) } (as expected). References Seress, A. Permutation Group Algorithms. Cambridge University Press, 2002. Lemmas in group theory