(January 2010) Let \(\sigma\) be the permutation \((4\ 2\ 1)(6\ 1\ 3\ 2)\) in \(S_6\). (a) Write \(\sigma\) as a product of disjoint cycles in \(S_6\). (b) Compute the order of \(\sigma\). (c) Is \(\sigma\) an even or an odd permutation?
Solution
Solution. (a) σ = (1 3)(2 6 4). (b) The order of σ is the lcm of the orders of each individual cycle in its decomposition into disjoint cycles. Since the order of an n-cycle is n, we obtain o(σ) = lcm(3, 2) = 6. (c)σ is a product of a 2-cycle and a 3-cycle. The 2-cycle is an odd permutation (since 2 is even), and the 3-cycle is even (since 3 is odd), so σ is even + odd = odd.