Schreier's lemma

Statement

Suppose H is a subgroup of G with generating set S, that is, G = \langle S\rangle.

Let R be a right transversal of H in G with the neutral element e in R. In other words, let R be a set containing exactly one element from each right coset of H in G.

For each g\in G, we define \overline{g} as the chosen representative of the coset Hg in the transversal R.

Then H is generated by the set :{rs(\overline{rs})^{-1}|r\in R, s\in S}.

Hence, in particular, Schreier's lemma implies that every subgroup of finite index of a finitely generated group is again finitely generated.{{citation

Example

The group \mathbb{Z}_3 = \mathbb{Z},/,3 \mathbb{Z} is cyclic. Via Cayley's theorem, \mathbb{Z}_3 is isomorphic to a subgroup of the symmetric group S_3. 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 that S_3 is generated by S = { s_1 = (1\ 2),, s_2 = (1\ 2\ 3) }.

\mathbb{Z}_3 has just two right cosets in S_3, namely \mathbb{Z}_3 and S_3 \setminus \mathbb{Z}_3 = { (1\ 2), (1\ 3), (2\ 3)}, so we select the right transversal R = { r_1 = e,, r_2 = (1\ 2) }, and we have : \begin{align} r_1s_1 &= (1\ 2), & \text{so} && \overline{r_1s_1} &= (1\ 2) \ r_1s_2 &= (1\ 2\ 3), & \text{so} && \overline{r_1s_2} &= e \ r_2s_1 &= e , & \text{so} && \overline{r_2s_1} &= e \ r_2s_2 &= (2\ 3), & \text{so} && \overline{r_2s_2} &= (1\ 2). \ \end{align}

Finally, : r_1s_1\left(\overline{r_1s_1}\right)^{-1} = e : r_1s_2\left(\overline{r_1s_2}\right)^{-1} = (1\ 2\ 3) : r_2s_1\left(\overline{r_2s_1}\right)^{-1} = e : r_2s_2\left(\overline{r_2s_2}\right)^{-1} = (1\ 2\ 3).

Thus, by Schreier's lemma, { e, (1\ 2\ 3) } generates \mathbb{Z}_3, but having the identity in the generating set is redundant, so it can be removed to obtain another generating set for \mathbb{Z}_3, { (1\ 2\ 3) }.

References

References

1. Seress, Ákos. (2002). "Permutation group algorithms". Cambridge University Press.
2. Johnson, David Lawrence. (1980). "Topics in the Theory of Group Presentations". Cambridge University Press.