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Math 332: Abstract Algebra Prof. Belk Jim Belk March 8, 2013 Homework 5 Solutions Problem 1. The following tables show the number of elements of S6 with each cycle structure: Cycle Structure Number e 1 (∗ ∗) 15 (∗ ∗ ∗) 40 (∗ ∗ ∗ ∗) 90 (∗ ∗ ∗ ∗ ∗) 144 (∗ ∗ ∗ ∗ ∗ ∗) 120 Cycle Structure Number (∗ ∗)(∗ ∗) 45 (∗ ∗ ∗)(∗ ∗) 120 (∗ ∗ ∗ ∗)(∗ ∗) 90 (∗ ∗ ∗)(∗ ∗ ∗) 40 (∗ ∗)(∗ ∗)(∗ ∗) 15 Problem 2. Let n ∈ N, and let G be a subgroup of Sn . If i ∈ {1, 2, . . . , n}, the stabilizer of i in in G is the set stabG (i) = {α ∈ G | α(i) = i}. Proposition. The stabilizer stabG (i) is a subgroup of G. Proof. Note first that stabG (i) is nonempty, since it contains the identity element e. Now, if α, β ∈ stabG (i), then α(i) = i and β(i) = i. Then (αβ)(i) = α(β(i)) = α(i) = i, so αβ ∈ stabG (i). Finally, if α ∈ stabG (i), then α(i) = i, so α−1 (i) = α−1 (α(i)) = i, and therefore α−1 ∈ stabG (i). Problem 3. Proposition. If n ≥ 2, then all of the 2-cycles in Sn are conjugate to one another. Proof. Let n ≥ 2, and let α, β ∈ Sn be 2-cycles. We must show that α and β are conjugate. There are three cases: either α and β are disjoint, or α and β share a number, or α = β. Suppose first that α and β are disjoint. Then α = (r s) and β = (t u) for some distinct r, s, t, u ∈ {1, . . . , n}. In this case, we have α = (r s) = (t r u s)(t u)(t s u r) = (t r u s) β (t r u s)−1 so α and β are conjugate. Now suppose that α and β share a number, say α = (r s) and β = (r t) for some distinct r, s, t ∈ {1, . . . , n}. Then α = (r s) = (s t)(r t)(s t) = (s t) β (s t)−1 so α and β are conjugate in this case as well. Finally, if α = β, then α and β are clearly conjugate, since α = eβe−1 . Problem 4. The perfect riffle shuffle is the permutation 1 2 3 · · · 26 27 28 29 · · · 52 σ = . 1 3 5 · · · 51 2 4 6 · · · 52 = (2 3 5 9 17 33 14 27)(4 7 13 25 49 46 40 28 )(6 11 21 41 30 8 15 29) (10 19 37 22 43 34 16 31)(12 23 45 38 24 47 42 32)(20 39 26 51 50 48 44 36)(18 35). Note that σ 8 is the identity permutation, so eight perfect riffle shuffles will return a deck of cards to its original state.