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| | 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 group|presentation]] of a [[subgroup]]. |
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| | ==Definition== |
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| | Suppose <math>H</math> is a [[subgroup]] of <math>G</math>, which is finitely generated with generating set <math>S</math>, that is, ''G'' = <math>\scriptstyle \langle S\rangle</math>. |
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| | Let <math>R</math> be a right [[Transversal (combinatorics)|transversal]] of <math>H</math> in <math>G</math>. In other words, <math>R</math> is (the image of) a [[Section (fiber bundle)|section]] of the quotient map <math>G \to H\backslash G</math>, where <math>H\backslash G</math> denotes the set of '''right cosets''' of <math>H</math> in <math>G</math>. |
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| | We make the definition that given <math>g</math>∈<math>G</math>, <math>\overline{g}</math> is the chosen representative in the transversal <math>R</math> of the coset <math>Hg</math>, that is, |
| | :<math>g\in H\overline{g}.</math> |
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| | Then <math>H</math> is generated by the set |
| | :<math>\{rs(\overline{rs})^{-1}|r\in R, s\in S\}</math> |
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| | == Example == |
| | Let us establish the evident fact that the group '''Z'''<sub>3</sub> = '''Z'''/3'''Z''' is indeed cyclic. Via [[Cayley's theorem]], '''Z'''<sub>3</sub> is a subgroup of the [[symmetric group]] ''S''<sub>3</sub>. Now, |
| | : <math>\Bbb{Z}_3=\{ e, (1\ 2\ 3), (1\ 3\ 2) \}</math> |
| | : <math>S_3= \{ e, (1\ 2), (1\ 3), (2\ 3), (1\ 2\ 3), (1\ 3\ 2) \}</math> |
| | where <math>e</math> is the identity permutation. Note ''S''<sub>3</sub> = <math>\scriptstyle\langle</math>{ ''s''<sub>1</sub>=(1 2), ''s''<sub>2</sub> = (1 2 3) }<math>\scriptstyle\rangle</math>. |
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| | '''Z'''<sub>3</sub> has just two cosets, '''Z'''<sub>3</sub> and ''S''<sub>3</sub> \ '''Z'''<sub>3</sub>, so we select the transversal { ''t''<sub>1</sub> = ''e'', ''t''<sub>2</sub>=(1 2) }, and we have |
| | : <math>\begin{matrix} |
| | t_1s_1 = (1\ 2),&\quad\text{so}\quad&\overline{t_1s_1} = (1\ 2)\\ |
| | t_1s_2 = (1\ 2\ 3) ,&\quad\text{so}\quad& \overline{t_1s_2} = e\\ |
| | t_2s_1 = e ,&\quad\text{so}\quad& \overline{t_2s_1} = e\\ |
| | t_2s_2 = (2\ 3) ,&\quad\text{so}\quad& \overline{t_2s_2} = (1\ 2). \\ |
| | \end{matrix}</math> |
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| | Finally, |
| | : <math>t_1s_1\overline{t_1s_1}^{-1} = e</math> |
| | : <math>t_1s_2\overline{t_1s_2}^{-1} = (1\ 2\ 3)</math> |
| | : <math>t_2s_1\overline{t_2s_1}^{-1} = e </math> |
| | : <math>t_2s_2\overline{t_2s_2}^{-1} = (1\ 2\ 3).</math> |
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| | Thus, by Schreier's subgroup lemma, { e, (1 2 3) } generates '''Z'''<sub>3</sub>, but having the identity in the generating set is redundant, so we can remove it to obtain another generating set for '''Z'''<sub>3</sub>, { (1 2 3) } (as expected). |
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| | ==References== |
| | * Seress, A. Permutation Group Algorithms. Cambridge University Press, 2002. |
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| | [[Category:Theorems in group theory]] |
| | [[Category:Lemmas]] |
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.
Definition
Suppose 
is a subgroup of 
, which is finitely generated with generating set 
, that is, G = 
.
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
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,
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).
References
- Seress, A. Permutation Group Algorithms. Cambridge University Press, 2002.