abstract algebra: a study guide for beginners - Northern Illinois ...

20 CHAPTER 2. FUNCTIONS
21. Prove that if τ ∈ Sn is a permutation with order m, then στσ −1 has order m, for any
permutation σ ∈ Sn.
22. Show that S10 has elements of order 10, 12, and 14, but not 11 or 13.
23. Let S be a set, and let X ⊆ S. Let G = {σ ∈ Sym(S) | σ(X) = X}. Prove that G is
a group of permutations.
24. Let G be a group of permutations, with G ⊆ Sym(S), for the set S. Let τ be a fixed
permutation in Sym(S). Prove that
is a group of permutations.
τGτ −1 = {σ ∈ Sym(S) | σ = τγτ −1 for some γ ∈ G}
MORE PROBLEMS: §2.3
25. Consider the following permutations in S7.
�
1
σ =
3
2
2
3
5
4
4
5
6
6
1
�
7
7
and τ =
Compute the following products.
†(a) στ (b) τσ †(c) στσ −1 (d) τστ −1
� 1 2 3 4 5 6 7
2 1 5 7 4 6 3
26.†Using the permutations σ and τ from Problem 25, write each of the permutations στ,
τσ, τ 2σ, σ−1 , στσ −1 , τστ −1 and τ −1στ as a product of disjoint cycles. Write σ and
τ as products of transpositions.
� �
1 2 3 4 5 6 7 8 9 10
27. Write
as a product of disjoint cycles and as a
3 4 10 5 7 8 2 6 9 1
product of transpositions. Construct its associated diagram, find its inverse, and find
its order.
28. Let σ = (3, 6, 8)(1, 9, 4, 3, 2, 7, 6, 8, 5)(2, 3, 9, 7) ∈ S9.
(a) Write σ as a product of disjoint cycles.
†(b) Is σ an even permutation or an odd permutation?
†(c) What is the order of σ in S9?
(d) Compute σ −1 in S9.
29. Let σ = (2, 3, 9, 6)(7, 3, 2, 5, 9)(1, 7)(4, 8, 7) ∈ S9.
(a) Write σ as a product of disjoint cycles.
†(b) Is σ an even permutation or an odd permutation?
†(c) What is the order of σ in S9?
(d) Compute σ −1 in S9.
30. Find a formula for the number of cycles of length m in Sn, for 2 ≤ m ≤ n.
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