Schreier's Subgroup Lemma

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. Let S be a set of generators for a group G . Let H be a subgroup of G with a right transversal R , such that
1 \in R .
For x \in G , define \bar{x} to be the unique element in
R \cap Hx .
Then H is generated by the set
\{ rs \bar{(rs)}^{-1} \mid r \in R, s \in S \} .

References

  • Seress, A. Permutation Group Algorithms. Cambridge University Press, 2002.

 

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