ALGEBRA PROBLEM SHEET 3 1. We can write the cycle (123·&mi...

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ALGEBRA PROBLEM SHEET 3 1. We can write the cycle (123···m) as the composition of transpositions (1m)◦···◦(13)◦(12).Use this fact to write the following permutations as compositions of transpositions and to determine whether they are even or odd. (1234567)∈S7(2658310)∈S10(1234567)◦(2658310)∈S10(1269)(310754)∈S10 2. Suppose that the permutation p consists of k disjoint cycles of lengths l1, l2, . . . , lk, where we include cycles of length 1. Prove that the parity of the permutation p is equal to the parity of the integerl1 + l2 + · · ·+lk−k. (Hint: This result generalises your calculations from problem 1.) 5. Carefully explain your answers to the following questions. a) Whatistheorderof(3,2,10)inthedirectproductZ5×Z8×Z12? b) What are the possible orders of permutations in A6? (Hint: Write down the possible “cycle types” of permutations in A6 — in other words, the lengths of the disjoint cycles — using the result from problem 2. Can you determine the order of a permutation from its cycle type?) 6. (a) Show that 84 × (a, b) = (0, 0) for all (a, b) ∈ Z12 × Z14, where we are using the notation 84×(a,b) = (a,b)+(a,b)+···+(a,b). 􏰄 􏰃􏰂 􏰅 84 times Hence, conclude that Z12 × Z14 is not isomorphic to Z168. 9. (a) Prove that in an abelian group, the product of two elements of finite order also has finite order.

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