Abstract Algebra

Problems taken from Chapter 5 - Permutation Groups.

Problems

1. Consider the cycle \(\sigma = \begin{pmatrix}1&5&3&2&4\end{pmatrix}\in S_5\). What is the cycle decomposition of \(\sigma^{-1}\)? In general, given a cycle \[ \sigma = \begin{pmatrix} a_1 & a_2 & a_3 & \cdots & a_k\end{pmatrix} \] what is the cycle decomposition of \(\sigma^{-1}\)?
2. Consider the permutation \(\sigma \in S_{13}\) given by \[ \sigma = \begin{pmatrix} 1&2&3&4&5&6&7&8&9&10&11&12&13\\[1ex] 12&13&3&1&11&9&5&10&6&4&7&8&2 \end{pmatrix}. \] Find the cycle decomposition of \(\sigma\) and \(\sigma^{-1}\).
3. If \(\sigma = \begin{pmatrix} a_1& a_2 & \cdots & a_m\end{pmatrix}\) is an \(m\)-cycle, show that \(|\sigma| = m\).
4. Problem 12
5. Problem 14