====== The Composition Of Permutations Is A Permutation ======
==== Problem ====
Prove theorem [[Theorem:The Composition Of Permutations Is A Permutation]].  
As some reminders, you may use the following facts that you proved in either Math 301 or Math 340. You may use these facts without proof.  
  * The composition of two injective functions is injective.
  * The composition of two surjective functions is surjective.
  * A function is a bijection if it is both injective (1 to 1) and surjective (onto). Hence, the composition of two bijective functions is a bijection. 
  * You might find induction helps you get from a composition of two bijective functions is bijective to the composition of $n$ bijective functions is bijective.  

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==== Remarks ====  
  * Make remarks with a list.

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==== $\LaTeX$ version ====
<file tex problem.the_composition_of_permutations_is_a_permutation.tex>
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% RequiredPackages: \usepackage{tikz}
% RequiredMacros: \DeclareMathOperator{\aut}{Aut} 
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\begin{problem}
Prove theorem [[Theorem:The Composition Of Permutations Is A Permutation]]).  
As some reminders, you may use the following facts that you proved in either Math 301 or Math 340. You may use these facts without proof.  
\begin{itemize}
\item The composition of two injective functions is injective.
\item The composition of two surjective functions is surjective.
\item A function is a bijection if it is both injective (1 to 1) and surjective (onto). Hence, the composition of two bijective functions is a bijection. 
\item You might find induction helps you get from a composition of two bijective functions is bijective to the composition of $n$ bijective functions is bijective. 
\\end{itemize}
end{problem}
</file>

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==== External links ====
  * [[wp>Dihedral group]]

{{tag>problem ben}}