====== Permutation Group Generated By $S$ ======
==== Problem ====
Let $X$ be a set.  Let $S$ be a collection of permutations of $X$.
  -Show that there is a permutation group that contains $S$.  
  -Let $\left<S\right>$ be the intersection of all permutation groups that contain $S$.  Show that $\left<S\right>$ is a permutation group.
  -Why is $\left<S\right>$ the smallest permutation group that contains $S$.  In other words, if $H$ is any other permutation group that contains $S$,  why must we have $\left<S\right>\subseteq H$.

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==== Remarks ====  
  * We could alternately define $\left<S\right>$ as the set of permutations of $X$ that we can express as a finite composition of elements in $S$ and inverses of elements in $S$. 

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==== $\LaTeX$ version ====
<file tex problem.permutation_group_generated_by_s.tex>
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\begin{problem}
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\end{problem}
</file>

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==== External links ====
  * [[wp>Generating set of a group]]

{{tag>problem ben needsreview}}