Question

4. You do not need to formally prove your answers to the following (a) Find the inverse of the cycle σ (al a2 am) .. Exp ress vour answer in cyclic notation (b) Find the order of a cycle ơ-a1 a2 . . . amje Sn. (c) Find a formula for the order of any permutation σ E Sn based on its disjoint cycle decomposition

Answer 1

(a) sigma = (a_1 a_2 - a_m) eqn sigma^-1 = (a_1 a_m a_m - 1 --- a_3 a_2) (b) Order of sigma = (a_1 a_2 --- a_m) eqn is m. i.e, order of a m - cycle is m. (c) Let sigma eqn. Let alpha_1, alpha_2, ----, alpha_n be m_i - cycles, i = 1, 2, ---, n, s.t. alpha_i's are disjoint and sigma = alpha_1 alpha_2 --- alpha_n. Now, order of alpha_i is m_i i.e_1 (alpha_i)^m_i = e = identity element of s_n. let m be the order of sigma. Then, m must be multiples of m_i's & m must minimum. So, m = lim (m_1, m_2, --- m_n).