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Homework 7 Math 332, Spring 2013 These problems must be written up in LATEX, and are due this Friday, March 22. 1. Let G = 1 a a∈Z . 0 1 (a) Prove that G is a subgroup of GL(2, R). (b) Prove that G is isomorphic to Z. 2. Let G, H, and K be groups, and let φ : G → H and ψ : H → K be isomorphisms. Prove that ψ ◦ φ is an isomorphism from G to K. 3. Prove that there exists a subgroup of GL(3, R) that is isomorphic to GL(2, R) 4. Let G = U (15), the multiplicative group of units modulo 15. (a) Find an automorphism σ of G such that σ(2) = 7 and σ(14) = 14. (b) Find an automorphism ρ of G such that ρ(2) = 7 and ρ(14) = 11. (c) List the eight automorphisms of G. (d) What is the isomorphism type of Aut(G)? Explain.