MATH 241 Homework 1, Fall 2011

Math 241, Introduction to Abstract Algebra – Fall 2011–2012 Course website: http://people.aub.edu.lb/∼kmakdisi/ Problem set 1, due Friday, October 7 at the beginning of class Exercises from Fraleigh: Section 0, exercises 1, 2, 3, 12. Section 1, exercises 22, 29, 32, 33. Section 2, exercises 8, 9, 23, 26. Section 3, exercise 33. Section 4, exercise 8. Additional Exercises (also required): Exercise A1.1: (Adapted from Jacobson) Let a ∈ R∗ and b ∈ R. Consider the function fa,b ∈ Fun(R, R) given by fa,b(x) = ax + b. a) Show that f is a bijection, and find its inverse function. b) Let G be the set of functions {fa,b|a ∈ R∗, b ∈ R}. Show that G is a group, where the group operation is composition of functions. (Thus G is a subgroup of Bij(R, R).) c) Bonus problem: Show that the group G is isomorphic to a subgroup of GL2(R). Look at, but do not hand in: Section 0, exercises 5–10, 14, 15, 29–32, 36. Section 1, exercise 34. Section 2, exercises 1–5, 14–16, 27–30. Section 3, exercises 3–7, 18, 19, 29–32.