Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.
Concept
Schreier's lemma
Summary
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.
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.
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).
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.