2.4. HOMOMORPHISMS AND ISOMORPHISMS 119 (c) Suppose that ˛ and ˇ are elements of S n and that ˇ D g˛g ± 1 for some g 2 S n . Show that when ˛ and ˇ are written as a product of disjoint cycles, they both have exactly the same number of cycles of each length. (For example, if ˛ 2 S 10 is a product of two 3–cycles, one 2–cycle, and four 1–cycles, then so is ˇ .) We say that ˛ and ˇ have the same cycle structure . (d) Conversely, suppose ˛ and ˇ are elements of S n and they have the same cycle structure. Show that there is an element g 2 S n such that ˇ D g˛g ± 1 . The result of this exercise is as follows: Two elements of S n are conjugate if, and only if, they have the same cycle structure. 2.4.15. Show that ± is the unique homomorphism from S n onto f 1; ± 1 g . Hint: Let ' W S n ±! f˙ 1 g be a homomorphism. If '..12// D ± 1 , show, using the results of Exercise 2.4.14 , that ' D ± . If '..12// D C 1 , show that ' is the trivial homomorphism, '.²/ D 1 for all ² . 2.4.16. For m < n , we can consider S m as a subgroup of S n . Namely, S m is the subgroup of S n that leaves ﬁxed the numbers from m C 1 to n . The parity of an element of S m can be computed in two ways: as an element of
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