Abstract Algebra IBL Wiki

====== 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}}