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+====== When Do Two Simple Shifts Span The Sameset ======
+==== Problem ====
+Consider the sets $H_{12}$ and $H_{15}$ of simple shift permutations on alphabets with 12 and 15 letters respectively.
+  - For each $k\in\{0,1,2,\ldots,11\}$, make a list of the elements in $H_{12}$ that are in $\text{span}(\{\phi_k\})$ and state the order of $\phi_k$ as an element of $H_{12}$.
+  - For each $k\in\{0,1,2,\ldots,14\}$, make a list of the elements in $H_{15}$ that are in $\text{span}(\{\phi_k\})$ and state the order of $\phi_k$ as an element of $H_{15}$.
+  - In general, if we are considering simple shift permutations in $H_n$, then when does $\text{span}(\{\phi_j\})=\text{span}(\{\phi_k\})$? Make a conjecture about when these two spans are equal.  Then check your conjecture against the list above.
+----
+==== Remarks ====
+  * Make remarks with a list.
+----
+==== $\LaTeX$ version ====
+<file tex problem.when_do_two_simple_shifts_span_the_sameset.tex>
+\begin{problem}
+Consider the sets $H_{12}$ and $H_{15}$ of simple shift permutations on alphabets with 12 and 15 letters respectively.
+\begin{enumerate}
+\item For each $k\in\{0,1,2,\ldots,11\}$, make a list of the elements in $H_{12}$ that are in $\text{span}(\{\phi_k\})$ and state the order of $\phi_k$ as an element of $H_{12}$.
+\item For each $k\in\{0,1,2,\ldots,14\}$, make a list of the elements in $H_{15}$ that are in $\text{span}(\{\phi_k\})$ and state the order of $\phi_k$ as an element of $H_{15}$.
+\item In general, if we are considering simple shift permutations in $H_n$, then when does $\text{span}(\{\phi_j\})=\text{span}(\{\phi_k\})$? Make a conjecture about when these two spans are equal.  Then check your conjecture against the list above.
+\end{enumerate}
+\end{problem}
+</file>
+----
+==== External links ====
+  * [[wp>Dihedral group]]
+{{tag>problem}}

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