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Homework 11 Math 332, Spring 2013 These problems must be written up in LATEX, and are due this Friday, May 10. 1. Let G = U (5) × U (5), and let N = {(1, 1), (2, 3), (3, 2), (4, 4)}. (a) List the elements of the four cosets of N in G. (b) Determine the isomorphism type of G/N . 2. Let G = GL(2, R), and let N be the following normal subgroup of G: a 0 N = a ∈ R and a 6= 0 . 0 a 1 −1 (a) Determine the order of N in G/N . 1 1 2 1 (b) Find an element A ∈ SL(2, R) such that AN = N. 5 7 3. Let G = Z8 × Z2 . (a) Find a four-element subgroup H of G such that G/H is isomorphic to Z2 × Z2 . (b) Find a four-element subgroup K of G such that G/K is isomorphic to Z4 . 4. Let R be the group of real numbers under addition, let Z be the group of integers under addition, and consider the quotient group R/Z. (a) What is the order of 2/5 + Z in R/Z? Explain. (b) The group R/Z has four elements of order eight. List them. (c) Give an example of an element of R/Z with infinite order. 5. Let G be a group, let H be a subgroup of G, and let N be a normal subgroup of G. Define N H = {nh | n ∈ N and h ∈ H}. (a) Prove that N H is a subgroup of G. (b) Prove that if H is normal in G, then N H is also normal in G.