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Chapter 3 Review: Solved problems

1. (a) What are the possibilities for the order of an element of Z13×? Explain your answer.
    (b) Show that Z13× is a cyclic group.
    Solution

2. Find all subgroups of Z11×, and give the lattice diagram which shows the inclusions between them.     Solution

3. Let G be the subgroup of GL3 (R) consisting of all matrices of the form such that a and b are in R.
Show that G is a subgroup of GL3 (R).     Solution

4. Show that the group G in Problem 3 is isomorphic to the direct product R × R.     Solution

5. List the cosets of the cyclic subgroup < 9 > in Z20×. Is the factor group Z20× / < 9 > cyclic?
Use the notation Z20× = { ± 1, ± 3, ± 7, ± 9 }.     Solution

6. Let G be the subgroup of GL2 (R) consisting of all matrices of the form , and let N be the subset of all matrices of the form .
    (a) Show that N is a subgroup of G, and that N is normal in G.
    (b) Show that the factor group G / N is isomorphic to the multiplicative group R×.
    Solution

7. Assume that the dihedral group of order 8 is written as

D4 = { e,a,a2,a3,b,ab,a2b,a3b },

where a4 = e, b2 = e, and b a = a3 b. Let N be the subgroup < a2 > = { e, a2 }.
    (a) Show by a direct computation that N is a normal subgroup of D4.
    (b) Is the factor group D4 / N a cyclic group?
    Solution

8. Let G = D8, using notation similar to Problem 7 above, and let N = { e,a2,a4,a6 }.
    (a) List all left cosets and all right cosets of N, and verify that N is a normal subgroup of G.
    (b) Show that the factor group G / N has order 4, but is not cyclic.
    Solution


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