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HOMEWORK SET 3- MATH 310 Due: Monday, 7 October 2024 1. This problem concerns the order of elements of the symmetric group S,,. (1) Show that |S,|=n!. (2) Show that {(12),(23),---,(n1 n)} generates S,. (3) Find the order of a cycle (i1ip...1;) S,. (4) Find the maximal possible order of an element o Sy,. 2. Consider the following subset of S : {e, (12)(34), (13)(24), (14)(23)}. (1) Fill in the multiplication table below. @E) aDE3) 1231 (13)(29) (1 4)(2 3) REMINDER : Our convention for the multiplication table is to write the multiplication in the order shown at right. Use the multiplication table to answer the following ques- tions. (2) Prove that this subset is a subgroup of Sy. (3) Is this subgroup abelian? (4) Is this subgroup cyclic? 3. (1) Let H be a subgroup of S,, which is not contained in A,,. Prove that |H|=2/HnA,,|. (2) Let 0,7 S, satisfy o7 =70. If 0 is an ncycle, prove that 7 = o* for some k. 4. (1) If o is a cycle of odd length, prove that o2 is also a cycle. (2) Let r, s be distinct elements of {1,2,---,n}. Show that A,(n > 3) is generated by the 3-cycles {(r s k)|[1