List two examples of nontrivial proper subgroups of the indicated group. a) Z18 b)U(18)
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- 18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.Let be a subgroup of a group with . Prove that if and only ifExercises 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