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High Quality Content by WIKIPEDIA articles! Schreier's subgroup lemma is a theorem in group theory used in the Schreier Sims algorithm and also for finding a presentation of a subgroup. Suppose H is a subgroup of G, which is finitely generated with generating set S, that is, G = . Let R be a right transversal of H in G. We make the definition that given g G, overline{g} is the chosen representative in the transversal R of the coset Hg, that is, gin Hoverline{g}. Then H is generated by the set {rs(overline{rs})^{-1} rin R, sin S}. 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, Bbb{Z}_3={ e, (1 2 3), (1 3 2) } S_3={ e, (1 2), (1 3), (2 3), (1 2 3), (1 3 2) } where e is the identity permutation. Note S3 = .