MATH 412 PROBLEM SET 8, DUE Wednesday, November 7, 2018 Reading: You should have read all of Chapter 7.1–7.5 by Monday November 5, as well as the Supplement on Group Actions. Problems due Wednesday before class starts A. Let S1 = fz 2 C j jzj = 1g ⊂ C. (1) Prove that S1 is an abelian group under multiplication, called the circle group. Draw a picture of S1 in the complex plane. (2) Prove that the map e : R ! C sending t 7! cos(2πt) + i sin(2πt) induces a surjective group homomorphism from the additive group (R; +) to the multiplicative group (S1; ×). φ (3) Prove that the kernel of e is Z, where by kernel of a group homomorphism G ! H we mean fg 2 G j φ(g) = eH g. B. (1) Prove that (Zm × Zn; +) is cyclic if and only if (m; n) = 1. You may use results from previous problem sets. (2) Give an example of two cyclic groups G and H such that G × H is cyclic. (3) Give an example of two cyclic groups G and H such that G × H is not cyclic. (4) Which of the following are cyclic groups? Justify. U11; U8; R; Z; Z × Z: T C. Recall that an n × n matrix A with real entries is orthogonal if AA = In. (1) Let On be the set of all n × n orthogonal matrices. Prove that On is a subgroup of GLn(R). (2) Let SLn(R) the set of all real n × n matrices of determinant 1. Prove that SLn is a subgroup of GLn(R). [Hint: Use facts from Math 217!] (3) Prove that in general, for any two subgroups H; K of a group G, the set H \ K is a subgroup. ∼ 1 1 (4) Prove that O2 \ SL2(R) = S , the circle group. (Hint: consider the map S ! GL2(R) sending a point (cos θ; sin θ) on the circle S1 to the matrix representing rotation through θ.) D. Let G = hgi be a cyclic group of order n. (1) Fix positive integers d; m with djm. Prove that hgmi ⊂ hgdi. (2) Now let d = gcd(m; n). Prove that hgmi = hgdi. (3) Prove that gm is a generator of G if and only of gcd(m; n) = 1. (4) For each of the cyclic groups G you found in B (4), list all elements g 2 G such that G = hgi. " p # 3 − 1 2 p 2 (5) Consider the element A = 1 3 2 O2, the group of orthogonal matrices, and let G be the 2 2 cyclic subgroup it generates. What is the order of G? Find all generators for G. [Hint: Interpret A geometrically!] 1.