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 .
Answer
(c) \(\tau \notin A_{6}\), because its sign is -1, which means it's an odd permutation. \(\boxed{\text{Final Answer}}\)
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. \(\boxed{\text{Final Answer}}\)
