Consider the sets $H_{12}$ and $H_{15}$ of simple shift permutations on alphabets with 12 and 15 letters respectively.

1. 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 $H_{12}$.
2. 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 $H_{15}$.
3. 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.

• Make remarks with a list.

problem.when_do_two_simple_shifts_span_the_sameset.tex

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\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}

• Dihedral group

problem