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--- problem:when_do_two_simple_shifts_span_the_sameset [2013/11/25 13:46]
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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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