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====== The Set Of Simple Shift Permutations======
==== Problem ====
Let $S = \{a,b,c,\ldots,z\}$ be the set of letters in the Roman alphabet. For each $n\in\mathbb{Z}$, let $\phi_{n}:S\to S$ represent the encryption key that shifts each letter in the alphabet right $n$, where we wrap around from $z$ to $a$ for letters that need to be shifted past $z$. So $\phi_3(a)=d$, $\phi_3(b)=3$, and $\phi_3(z)=c$. We've called this a simple shift permutation of $S$.

  - For which $n$ does $\phi_n$ not change the message?
  - Consider the encryption key $\phi_{9}$, which shifts each letter right 9.  If a message had been encrypted using $\phi_9$, then clearly $\phi_{-9}$ would decrypt the message as $\phi_{-9}(\phi_9(s))=s$ for any letter $s$. Give a positive integer $n$ that we could also use to decode a message that has been encrypted using $\phi_9$.
  - Does $\phi_{30}=\phi_7$?  Does $\phi_{33}=\phi_7$?  For which $n$ does $\phi_n=\phi_7$? Explain.
  - Consider the set of all shifts of $S$, namely $H=\{\phi_n\mid n\in\mathbb{Z}\}$.  How many different functions are in $H$? 

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

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==== $\LaTeX$ version ====
<file tex problem.the_set_of_simple_shift.tex>
%%%%%
% DEPENDENCIES
% RequiredPackages: \usepackage{tikz}
% RequiredMacros: \DeclareMathOperator{\aut}{Aut} 
%%%%%
\begin{problem}
Let $S = \{a,b,c,\ldots,z\}$ be the set of letters in the Roman alphabet. For each $n\in\mathbb{Z}$, let $\phi_{n}:S\to S$ represent the encryption key that shifts each letter in the alphabet right $n$, where we wrap around from $z$ to $a$ for letters that need to be shifted past $z$. So $\phi_3(a)=d$, $\phi_3(b)=3$, and $\phi_3(z)=c$. We've called this a simple shift permutation of $S$.

\begin{enumerate}
\item For which $n$ does $\phi_n$ not change the message?
\item Consider the encryption key $\phi_{9}$, which shifts each letter right 9.  If a message had been encrypted using $\phi_9$, then clearly $\phi_{-9}$ would decrypt the message as $\phi_{-9}(\phi_9(s))=s$ for any letter $s$. Give a positive integer $n$ that we could also use to decode a message that has been encrypted using $\phi_9$.
\item Does $\phi_{30}=\phi_7$?  Does $\phi_{33}=\phi_7$?  For which $n$ does $\phi_n=\phi_7$? Explain.
\item Consider the set of all shifts of $S$, namely $H=\{\phi_n\mid n\in\mathbb{Z}\}$.  How many different functions are in $H$?
\end{enumerate}
\end{problem}
</file>

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

{{tag>problem ben}}
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