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Problem (When Do Two Simple Shifts Span The Same Set)
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.
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