Mathematics 1214: Introduction to Group Theory
Tutorial exercise sheet 3
−1
1 2 3 4 5 6 7
1. (a) Compute
, writing your answer in 2-row notation
3 1 7 6 5 4 2
and as a product of disjoint cycles. Is this an even or an odd permutation?
(b) Let n ∈ N, and consider a permutation
1
2
...
α=
α(1) α(2) . . .
n
.
α(n)
Explain how to find α−1 in 2-row notation.
2. A deck of 52 playing cards is shuffled repeatedly using the following procedure:
Swap the top two cards, then cut the deck by moving the top 26 cards to the bottom.
How many of these shuffles must be performed before the cards are back in their
original order?
3. Let n, k ∈ N, let a1 , a2 , . . . , ak be distinct elements of {1, 2, . . . , n}, and consider
the cycle α = (a1 a2 . . . ak ) ∈ Sn . Prove that if β ∈ Sn then
β ◦ α ◦ β −1 = (b1 b2 . . . bk )
where bi = β(ai ) for 1 ≤ i ≤ k.
4. Let n ∈ N.
(a) Show that if α ∈ Sn , then α is an even permutation if and only if α−1 is an
even permutation.
(b) Show that if α, β ∈ Sn then σ(β ◦ α ◦ β −1 ) = σ(α).
5. Let n ∈ N,
(a) Let An = {α ∈ Sn : α is an even permutation}. Prove that (An , composition)
is a group. [Hint: use Exercise 4(a).]
(b) Let Bn = {β ∈ Sn : β is an odd permutation}. Is Bn a group under composition?
(c) Suppose that n ≥ 2. Show that the mapping
T : An → Bn ,
α 7→ α ◦ (1 2)
is a bijection. [Along the way, you should prove that T is well-defined by
checking that if α ∈ An then T (α) ∈ Bn .]
(d) Deduce that if n ≥ 2, then |An | = |Bn |, and hence compute the order of An .
What happens when n = 1?
6. Let n, k ∈ N with k ≥ 2, let a1 , a2 , . . . , ak be distinct elements of {1, 2, . . . , n} and
let α be the cycle in Sn given by α = (a1 a2 . . . ak ). Recall that α2 is defined to
be α ◦ α. Show that if k is even then α2 is the product of two disjoint cycles.
What happens if k is odd? [Justify your answer].