Chapter 4.1, Problem 3E

In each part of Exercise $1$, decide whether the permutation is even or odd.

Express each permutation as a product of disjoint cycles and find the orbits of each permutation.

a. $\left[\begin{array}{ccccc}1& 2& 3& 4& 5\\ 4& 5& 3& 1& 2\end{array}\right]$

b. $\left[\begin{array}{ccccc}1& 2& 3& 4& 5\\ 1& 3& 2& 5& 4\end{array}\right]$

c. $\left[\begin{array}{ccccc}1& 2& 3& 4& 5\\ 4& 1& 3& 5& 2\end{array}\right]$

d. $\left[\begin{array}{ccccc}1& 2& 3& 4& 5\\ 3& 5& 2& 4& 1\end{array}\right]$

e. $\left[\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 3& 4& 5& 6& 1& 2& 7\end{array}\right]$

f. $\left[\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 5& 1& 3& 7& 2& 6& 4\end{array}\right]$

g. $\left[\begin{array}{ccccc}1& 2& 3& 4& 5\\ 1& 3& 4& 5& 2\end{array}\right]\left[\begin{array}{ccccc}1& 2& 3& 4& 5\\ 3& 2& 4& 1& 5\end{array}\right]$

h. $\left[\begin{array}{ccccc}1& 2& 3& 4& 5\\ 2& 3& 4& 1& 5\end{array}\right]\left[\begin{array}{ccccc}1& 2& 3& 4& 5\\ 1& 3& 5& 4& 2\end{array}\right]$