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.