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 = {z ∈ C | |z| = 1} ⊂ 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 {g ∈ G | φ(g) = eH }.

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 d|m. 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 ∈ G such that G = hgi. " √ # 3 − 1 2 √ 2 (5) Consider the element A = 1 3 ∈ 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!]

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