16 Chapter 1. Groups I (ii) Compute the parity of f. (iii) Compute the inverse of f. 1.19. If α is an r-cycle and 1 k r, is αk an r-cycle? ∗ 1.20. (i) Prove that if α and β are (not necessarily disjoint) permutations that commute, then (αβ)k = αkβk for all k ≥ 1. Hint. First show that βαk = αkβ by induction on k. (ii) Give an example of two permutations α and β for which (αβ)2 = α2β2. ∗ 1.21. (i) Prove, for all i, that α ∈ Sn moves i if and only if α−1 moves i. (ii) Prove that if α, β ∈ Sn are disjoint and if αβ = (1), then α = (1) and β = (1). ∗ 1.22. Prove that the number of even permutations in Sn is 1 2 n!. Hint. Let τ = (1 2), and define f : An → On, where An is the set of all even permutations in Sn and On is the set of all odd permutations, by f : α → τα. Show that f is a bijection, so that |An| = |On| and, hence, |An| = 1 2 n!. ∗ 1.23. (i) How many permutations in S5 commute with α = (1 2 3), and how many even permutations in S5 commute with α? Hint. Of the six permutations in S5 commuting with α, only three are even. (ii) Same questions for (1 2)(3 4). Hint. Of the eight permutations in S4 commuting with (1 2)(3 4), only four are even. 1.24. Give an example of α, β, γ ∈ S5, with α = (1), such that αβ = βα, αγ = γα, and βγ = γβ. ∗ 1.25. If n ≥ 3, prove that if α ∈ Sn commutes with every β ∈ Sn, then α = (1). 1.26. If α = β1 · · · βm is a product of disjoint cycles and δ is disjoint from α, show that β11 e · · · βm em δ commutes with α, where ej ≥ 0 for all j. Section 1.3. Groups Since Galois’s time, groups have arisen in many areas of Mathematics other than the study of roots of polynomials, for they are the precise way to describe the notion of symmetry, as we shall see. The essence of a “product” is that two things are combined to form a third thing of the same kind. For example, ordinary multiplication, addition, and subtraction combine two numbers to give another number, while composition combines two permutations to give another permutation. Definition. A binary operation on a set G is a function ∗ : G × G → G. In more detail, a binary operation assigns an element ∗(x, y) in G to each ordered pair (x, y) of elements in G. It is more natural to write x ∗ y instead of ∗(x, y) thus, composition of functions is the function (f, g) → g ◦ f multiplication,

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