-
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Ym)-1
YmYm-1...7271.
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-1
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20. Show that (1 2) is not a produ
21. (a) If 7. 12....m are transpositions, show that
(
(b) Show that o and 0-1 have the same parity for all o in Sn.
(c) Show that o and Tot-1 have the same parity for all o and T in Sn:
33. Show that Antun Sn = A, for all n 2 3 (regard Sm C Sn+1 in the usual way).
23. Leto € Sm, o te. If n 23, show that y € Sn exists such that or + yo. (Hint: If
ok=1 with k #1, choose m€ {k,1; and take y = (k m).]
24. If o E Sn, show that o2 = e if and only if o is a product of disjoint transpositions.
25. If n23, show that every even permutation in Sn is a product of 3-cycles.
26. Let y be any cycle of length r. If o e Sn, show that oyo-1 is also a cycle of length r.
More precisely, if y = (kı k2
okr).
kr) show that oyo-1 = (oki ok2
27. (a) Show that (kı kz ... kr) = (ki kr) (kı kr-1)... (kı k2).
k2
= (-
(b) Show that each o ESn is a product of the transpositions (1 2), (1 3),...,(1 n).
(Hint: Each transposition is such a product by (a) and Exercise 26.]
(c) Repeat (b) for the transpositions (1 2), (2 3),..., (n-1 n). (Hint: Use (a)
and Exercise 26.)
(d) If o= (1 2 3 n), show that each element of Sn is a product of the
permutations (1 2), o, and 0-1. (Hint: Use (b) and Exercise 26.]
28. Let o = (1 2 3 ... n) be a cycle of length n > 2.
(a) If n = 2k, find the factorization of o2 into disjoint cycles.
9
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