In group theory, Schreier's lemma is a theorem used in the SchreierâÂÂSims algorithm and also for finding a presentation of a subgroup.
Suppose is a subgroup of with generating set , that is, .
Let be a right transversal of in with the neutral element in . In other words, let be a set containing exactly one element from each right coset of in .
For each , we define as the chosen representative of the coset in the transversal .
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.
The group is cyclic. Via Cayley's theorem, is isomorphic to a subgroup of the symmetric group . Now,
where is the identity permutation. Note that is generated by .
has just two right cosets in , namely and , so we select the right transversal , and we have
Finally,
Thus, by Schreier's lemma, generates , but having the identity in the generating set is redundant, so it can be removed to obtain another generating set for , .