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Extra info for Algebra: Monomials and Polynomials

Sample text

58. 52 the matrix 0 −1 −1 0 A= . Express A as a power of the other non-identity matrices of the group. 59. 36 you showed that the quaternions form a group under matrix multiplication. Verify that H = {1, −1, i, −i} is a cyclic group. What elements generate H ? 60. 54(C). 61. Let G be a group, and g ∈ G. Let d , n ∈ Z and assume ord ( g ) = d . Show that g n = e if and only if d | n. 62. Show that any group of 3 elements is cyclic. 63. 32 on page 30) cyclic? What about the cyclic group of order 4?

To do that, it is helpful to observe two important properties. 2. 41. In D3 , ϕρ = ρ2 ϕ. Proof. Compare ϕρ = −1 0 0 1 3 2 − 12 − 12 − 3 2 = 1 2 3 2 3 2 − 12 and ρ2 ϕ = = = − 12 − 3 2 3 − 12 2 − 12 23 − 23 − 12 1 3 2 2 3 − 12 2 3 2 − 12 − 12 − 3 2 −1 0 0 1 −1 0 0 1 . 41? It implies that multiplication in D3 is non-commutative! We have ϕρ = ρ2 ϕ, and a little logic (or an explicit computation) shows that ρ2 ϕ = ρϕ: thus ϕρ = ρϕ. 42. In D3 , ρ3 = ϕ 2 = ι. Proof. You do it! 43. Exercises. 43. Show explicitly (by matrix multiplication) that in D3 , ρ3 = ϕ 2 = ι.

Admittedly, the two are not identical: Mn is the set of products of powers of n distinct variables, whereas Mn is a set of lists of powers of one variable. In addition, if the variables are not commutative (remember that this can occur), then Mn and Mn are not at all similar. Think about ( x y )4 = xy xy xy xy; if the variables are commutative, we can combine them into x 4 y 4 , which looks likes (4, 4). If the variables are not commutative, however, it is not at all clear how we could get ( xy )4 to correspond to an element of N × N.

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