4. Consider the subgroup H = {1, 10, 26} of U37. (a) Find and compare the orders of 6 E U37 and 6H E U37/H. (b) Repeat part (a) with 34 E U37 and 34H E U37/H. It might help to write 34 as -3 modulo 37. (c) Repeat part (a) with 4 E U37 and 4H E U37/H. (d) What conjecture do you have?

9. Find all homomorphic images of the octic 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.

Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.

11. Find all normal subgroups of the alternating group .

12. Find all normal subgroups of the quaternion group.

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

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

Exercises 1. List all cyclic subgroups of the group in Example of section. Example 3. We shall take and obtain an explicit example of . In order to define an element of , we need to specify , , and . There are three possible choices for . Since is to be bijective, there are two choices for after has been designated, and then only one choice for . Hence there are different mappings in .

Exercises In Section 3.3, the centralizer of an element a in the group G was shown to be the subgroup given by Ca=xGax=xa. Use the multiplication table constructed in Exercise 20 to find the centralizer Ca for each element a of the octic group D4. Construct a multiplication table for the octic group D4 described in Example 12 of this section.