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.
, which is finitely generated with generating set
be a right transversal of
is (the image of) a section of the quotient map
denotes the set of right cosets of
The definition is made given that
is the chosen representative in the transversal
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.
The group Z3 = Z/3Z is cyclic.
Via Cayley's theorem, Z3 is a subgroup of the symmetric group S3.
is the identity permutation.
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 it can be removed to obtain another generating set for Z3, { (1 2 3) } (as expected).