Exercise 7
Let $τ = (\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6 \\ 6 & 4 & 3 & 5 & 2 & 1 \end{array}), σ = (\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \\ 5 & 4 & 2 & 1 & 3 \end{array})$
(a) Compute each of the following:
(a $τ^{0}$
(2) $τ^{3}$
(3) $σ^{- 3}$
(b) Compute each of the following:
(1) $| σ |$
(2) $sgn (τ)$
(3) $sgn (σ τ)$
(c) Does $τ \in A_{6}$ ? Justify your answer .

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Step 1 ：(a) (1) $τ^{0} = (1, 6, 4, 5, 2, 3)$ (2) $τ^{3} = (1, 6, 4, 5, 2, 3)$ (3) $σ^{- 3} = (1, 4, 3, 5, 2)$

Step 2 ：(b) (1) $| σ | = 1$ (2) $sgn (τ) = - 1$ (3) $sgn (σ τ) = 1$

Step 3 ：(c) $τ \notin A_{6}$ , because its sign is -1, which means it's an odd permutation. $Final Answer$