2. List all of the elements in each of the following subgroups (a) The subgroup of Z24 generated by 15. (b) The subgroup generated by 3 in U (20).

4. List all the elements of the subgroupin the group under addition, and state its order.

Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.

For each of the following subgroups H of the addition groups Z18, find the distinct left cosets of H in Z18, partition Z18 into left cosets of H, and state the index [ Z18:H ] of H in Z18. H= [ 8 ] .

3. Consider the group under addition. List all the elements of the subgroup, and state its order.

Exercises 13. For each of the following values of, find all subgroups of the group described in Exercise, addition and state their order. a. b. c. d. e. f.

11. Find all normal subgroups of the alternating group .

34. Suppose that and are subgroups of the group . Prove that is a subgroup of .

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

Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.