2. Let a be an element of a group such that lal = 15. Find all the right cosets of the cyclic subgroup generated by (as) in the cyclic subgroup generated by
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- 11. Find all normal subgroups of the alternating group .27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.Exercises 10. Find an isomorphism from the multiplicative group to the group with multiplication table in Figure . This group is known as the Klein four group. Figure Sec. 16. a. Prove that each of the following sets is a subgroup of , the general linear group of order over . Sec. 3. Let be the Klein four group with its multiplication table given in Figure . Figure Sec. 17. Show that a group of order either is cyclic or is isomorphic to the Klein four group . Sec. 16. Repeat Exercise with the quaternion group , the Klein four group , and defined by
- 4. List all the elements of the subgroupin the group under addition, and state its order.Consider the group U9 of all units in 9. Given that U9 is a cyclic group under multiplication, find all subgroups of U9.Exercises 8. Find an isomorphism from the group in Example of this section to the multiplicative group . Sec. 16. Prove that each of the following sets is a subgroup of , the general linear group of order over .