Hamilton–Jacobi equation: Difference between revisions

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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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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>&isin;<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,
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:<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 ==
</ul>
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>.
 
'''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>
 
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>
 
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).
 
==References==
* Seress, A. Permutation Group Algorithms. Cambridge University Press, 2002.
 
[[Category:Theorems in group theory]]
[[Category:Lemmas]]

Latest revision as of 19:01, 30 October 2014

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