Chapter 9

Let \(\pi\) and \(\rho\) be the permutations $$ \pi=\left(\begin{array}{llllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 3 & 4 & 5 & 6 & 7 & 2 & 1 & 8 \end{array}\right) \text { and } \rho=\left(\begin{array}{llllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \end{array}\right) $$ Write \(\pi\) and \(\rho\) in cycle notation. Calculate \(\pi \rho, \rho \pi\) and \(\pi^{2} \rho\). Find the orders ar signs of \(\pi, \rho\) and \(\pi \rho\).

List all 24 elements of \(S(4)\) in cycle notation. Show that the subset. \(V=\left\\{1,\left(\begin{array}{lll}1 & 2\end{array}\right)(34),(13)(24),(14)(23)\right\\}\) is a subgroup of \(S(4)\). Use the fact that any permutation \(\pi\) has the same cycle structure as \(\rho \pi \rho^{-1}\) (for any \(\rho\) ) to show that \(V\) is a normal subgroup of \(S(4)\). Give names to the six left cosets of \(V\), and write down the table for the quotient group \(S(4) / V\).