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Math 366 Assignment 2 Due Monday, October 25 1. Suppose H is a proper subgroup of Z (under addition) and that 18, 30, 40 ∈ H. Which subgroup of Z is H? 2. Suppose a group G is defined by the following Cayley table: 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 2 2 1 4 3 6 5 8 7 3 3 8 5 2 7 4 1 6 4 4 7 6 1 8 3 2 5 5 5 6 7 8 1 2 3 4 6 6 5 8 7 2 1 4 3 0 1 −1 0 7 7 4 1 6 3 8 5 2 8 8 3 2 5 4 7 6 1 (a) Find the centralizer of each element of G. (b) Find Z(G). (c) Find the order of each element of G. 3. Prove that the center of a group is Abelian. 4. Consider the elements A= and B= 0 −1 1 −1 of SL(2, R). Find the orders of A, B, and AB. 5. Find all generators of Z6 and Z8 (both under addition!). 6. Write out the elements of the subgroups h3i and h7i in U (20). 7. Find a collection of distinct subgroups ha1 i, ha2 i, . . . , han i of Z120 , each a proper subgroup of the next, with n as large as possible. (Hint: What subgroups could there be? How could they fit into chains like this? Think about divisibility of subgroup orders....) 8. Find the order of each of the following permutations: (a) (14) (b) (147) (c) (124)(357) (d) (124)(35) (e) (124)(357869) NOTE: Much of this is from the 6th edition of Gallian’s Contemporary Abstract Algebra. 1