Prove part (A) by filling in the blanks. Let σ ∈ SX be an arbitrary cycle of length k. Then σ can be written as (a0 a1 . . . ak−1), for some set of elements a0, a1, . . . ak−1 in X. In order to show that σ k = id, it is sufficient to show that σ k (x) = < 1 > ∀x ∈ X. Let A be the set {a0, a1, . . . ak−1}. Now for any x ∈ X, there are two possibilities: (i) x ∈ X\A; (ii) x ∈ A. We’ll deal with these two cases separately. (i) In this case, σ(x) = < 2 > It follows that σ 2 (x) = σ(σ(x)) = σ( < 3 > ) = < 4 > . We can use the same argument to show that σ 3 (x) = < 5 > , and that σ k (x) = < 6 > for any natural number < 7 > . (ii) In this case, then x = aj for some integer j, 1 ≤ j ≤ < 8 > . It follows from the definition of cycle that σ(x) = σ(aj ) = a mod (j+1,k) . Furthermore, σ 2 (x) = σ(a mod (j+1,k) ) = < 9 > . Similarly it follows that σ k (x) = a mod (j+ <10> ,k) = a <11> = x. Cases (i) and (ii) establish that ∀x ∈ X, < 12 > = x. It follows that σ k = < 13 > . (Blanks are the spots with numbers in between these symbols < >)
Prove part (A) by filling in the blanks.
Let σ ∈ SX be an arbitrary cycle of length k. Then σ can be written as (a0 a1 . . . ak−1), for some set of elements a0, a1, . . . ak−1 in X. In order to show that σ k = id, it is sufficient to show that σ k (x) = < 1 > ∀x ∈ X. Let A be the set {a0, a1, . . . ak−1}. Now for any x ∈ X, there are two possibilities: (i) x ∈ X\A; (ii) x ∈ A. We’ll deal with these two cases separately.
(i) In this case, σ(x) = < 2 > It follows that σ 2 (x) = σ(σ(x)) = σ( < 3 > ) = < 4 > . We can use the same argument to show that σ 3 (x) = < 5 > , and that σ k (x) = < 6 > for any natural number < 7 > .
(ii) In this case, then x = aj for some integer j, 1 ≤ j ≤ < 8 > . It follows from the definition of cycle that σ(x) = σ(aj ) = a mod (j+1,k) . Furthermore, σ 2 (x) = σ(a mod (j+1,k) ) = < 9 > . Similarly it follows that σ k (x) = a mod (j+ <10> ,k) = a <11> = x.
Cases (i) and (ii) establish that ∀x ∈ X, < 12 > = x.
It follows that σ k = < 13 > .
(Blanks are the spots with numbers in between these symbols < >)
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