# Schreier's lemma

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 $H$ is a subgroup of $G$ , which is finitely generated with generating set $S$ , that is, G = $\langle S\rangle$ .

Let $R$ be a right transversal of $H$ in $G$ . In other words, $R$ is (the image of) a section of the quotient map $G\to H\backslash G$ , where $H\backslash G$ denotes the set of right cosets 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,

$g\in H{\overline {g}}.$ Then $H$ is generated by the set

$\{rs({\overline {rs}})^{-1}|r\in R,s\in S\}$ ## 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,

$\mathbb {Z} _{3}=\{e,(1\ 2\ 3),(1\ 3\ 2)\}$ $S_{3}=\{e,(1\ 2),(1\ 3),(2\ 3),(1\ 2\ 3),(1\ 3\ 2)\}$ where $e$ is the identity permutation. Note S3 = $\langle$ { s1=(1 2), s2 = (1 2 3) }$\rangle$ .

Z3 has just two cosets, Z3 and S3 \ Z3, so we select the transversal { t1 = e, t2=(1 2) }, and we have

${\begin{matrix}t_{1}s_{1}=(1\ 2),&\quad {\text{so}}\quad &{\overline {t_{1}s_{1}}}=(1\ 2)\\t_{1}s_{2}=(1\ 2\ 3),&\quad {\text{so}}\quad &{\overline {t_{1}s_{2}}}=e\\t_{2}s_{1}=e,&\quad {\text{so}}\quad &{\overline {t_{2}s_{1}}}=e\\t_{2}s_{2}=(2\ 3),&\quad {\text{so}}\quad &{\overline {t_{2}s_{2}}}=(1\ 2).\\\end{matrix}}$ Finally,

$t_{1}s_{1}{\overline {t_{1}s_{1}}}^{-1}=e$ $t_{1}s_{2}{\overline {t_{1}s_{2}}}^{-1}=(1\ 2\ 3)$ $t_{2}s_{1}{\overline {t_{2}s_{1}}}^{-1}=e$ $t_{2}s_{2}{\overline {t_{2}s_{2}}}^{-1}=(1\ 2\ 3).$ 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).