====== The Span Of A Simple Shift ======
==== Problem ====
Suppose that our alphabet $S$ consists of only 12 letters $S=\{a,b,c,d,e,f,g,h,i,j,k,l\}$. Let $H_{12}=\{\phi_n\mid n\in \mathbb{Z}\}$ be the set of simple shift permutations on this 12 letter alphabet (we wrap around from $l$ to $a$). 
  - For each $n\in \{0,1,2,\ldots,11\}$, list the elements in $\text{span}(\{\phi_n\} ) $.
  - For which $n$ does $\text{span}(\{\phi_n\})=H_{12}$. 
  - Make a conjecture about any patterns you see above and their relation to the size (12) of the alphabet. When you report your preparation, please tell me your conjecture.
  - Whenever you make a conjecture, you should always test your conjecture on an example you have not yet considered.  Pick another integer $k\neq 12,26$, and look at the set $H_k$ of simple shift permutations of an alphabet consisting of $k$ letters. Then check if your conjecture holds. 
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==== Remarks ====  
  * Make remarks with a list.

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==== $\LaTeX$ version ====
<file tex problem.the_span_of_a_simple_shift.tex>
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\begin{problem}
Suppose that our alphabet $S$ consists of only 12 letters $S=\{a,b,c,d,e,f,g,h,i,j,k,l\}$. Let $H_{12}=\{\phi_n\mid n\in \mathbb{Z}\}$ be the set of simple shift permutations on this 12 letter alphabet (we wrap around from $l$ to $a$). 
\begin{enumerate}
\item For each $n\in \{0,1,2,\ldots,11\}$, list the elements in $\text{span}(\{\phi_n\} ) $.
\item For which $n$ does $\text{span}(\{\phi_n\})=H_{12}$. 
\item Make a conjecture about any patterns you see above and their relation to the size (12) of the alphabet. When you report your preparation, please tell me your conjecture.
\item Whenever you make a conjecture, you should always test your conjecture on an example you have not yet considered.  Pick another integer $k\neq 12,26$, and look at the set $H_k$ of simple shift permutations of an alphabet consisting of $k$ letters. Then check if your conjecture holds. 
\end{enumerate}
\end{problem}
</file>

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

{{tag>problem ben}}