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22. 23. 24. 25. 26. 27. 28. 29. 30. 31. letter to the one directly below it in the display following. Write o' in cycle form. ABCDEFGH I] KLMNOPQRS T UVWXYZ HDBGJ ECMILONP FKRUS AWQTVZXY If [r and B are distinct 2-cyc1es, what are the possibilities for Inl? Show that if H is a subgroup of Sm then either every member of H is an even permutation or exactly half of the members are even. (This exercise is referred to in Chapter 25.) Suppose that H is a subgroup of Sn of odd order. Prove that H is a subgroup of A\". Give two reasons why the set of odd permutations in Sn is not a subgroup. Let o: and [3 belong to S\". Prove that {FIE1&3 is an even permutation. Use Table 5.1 to compute the following. a. The centralizer of ['13 : (13)(24) b. The centralizer of u : (124) How many elements of order 5 are in 5?? How many elements of order 4 does 35 have? How many elements of order 2 does 86 have? Prove that (1234) is not the product of 3-cycles. Let B E 57 and suppose B4 2 (2143567). Find [3. What are the possibilities for [3 if )3 E SQ? 1- 1- rm i 1! I II . 1 a