Problem
Exercise 7 Let $\tau=\left(\begin{array}{llllll}1 & 2 & 3 & 4 & 5 & 6 \\ 6 & 4 & 3 & 5 & 2 & 1\end{array}\right), \sigma=\left(\begin{array}{lllll}1 & 2 & 3 & 4 & 5 \\ 5 & 4 & 2 & 1 & 3\end{array}\right)$ (a) Compute each of the following: (a $\tau^{0}$ (2) $\tau^{3}$ (3) $\sigma^{-3}$ (b) Compute each of the following: (1) $|\sigma|$ (2) $\operatorname{sgn}(\tau)$ (3) $\operatorname{sgn}(\sigma \tau)$ (c) Does $\tau \in A_{6}$ ? Justify your answer .
Solution
Step 1 ：(a) (1) \(\tau^{0} = (1, 6, 4, 5, 2, 3)\) (2) \(\tau^{3} = (1, 6, 4, 5, 2, 3)\) (3) \(\sigma^{-3} = (1, 4, 3, 5, 2)\)
Step 2 ：(b) (1) \(|\sigma| = 1\) (2) \(\operatorname{sgn}(\tau) = -1\) (3) \(\operatorname{sgn}(\sigma \tau) = 1\)
Step 3 ：(c) \(\tau \notin A_{6}\), because its sign is -1, which means it's an odd permutation. \(\boxed{\text{Final Answer}}\)
