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Homework 8 Math 332, Spring 2013 These problems must be written up in LATEX, and are due this Friday, April 5. 0 2 1. (a) Find an eight-element subgroup G of GL(2, Z3 ) containing the matrices 1 0 1 1 and . 1 2 (b) What is the isomorphism type of G? 2. Let T1 : R3 → R3 be a 180◦ rotation about the line y = −x in the xy-plane, and let T2 : R3 → R3 be a 120◦ rotation about the line x = y = z (either rotation will do). (a) Find 3 × 3 matrices A1 and A2 corresponding to the linear transformations T1 and T2 . (b) Find a six-element subgroup G of GL(3, R) containing A1 and A2 . (c) What is the isomorphism type of G? 3. Let G be the group of symmetries of the following (flattened) tetrahedron. H–1, 1, –18L H1, 1, 18L H–1, –1, 18L H1, –1, –18L (a) List the eight 3 × 3 matrices corresponding to the eight elements of G. (b) What is the isomorphism type of G?