Abstract Algebra

Problems taken from Chapter 5 - Permutation Groups.

Problems

1. Consider the cycle $\sigma =\left(\begin{array}{ccccc}1& 5& 3& 2& 4\end{array}\right)\in S_5$. What is the cycle decomposition of ${\sigma }^{-1}$? In general, given a cycle $$\sigma =\left(\begin{array}{ccccc}{a}_1& a_2& a_3& \cdots & a_k\end{array}\right)$$ what is the cycle decomposition of ${\sigma }^{-1}$?
2. Consider the permutation $\sigma \in S_{13}$ given by $$\sigma =\left(\begin{array}{ccccccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13\\ 12& 13& 3& 1& 11& 9& 5& 10& 6& 4& 7& 8& 2\end{array}\right).$$ Find the cycle decomposition of $\sigma$ and ${\sigma }^{-1}$.
3. If $\sigma =\left(\begin{array}{cccc}{a}_1& a_2& \cdots & a_m\end{array}\right)$ is an $m$-cycle, show that $|\sigma |=m$.
4. Problem 12
5. Problem 14