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- 11. Find all normal subgroups of the alternating group .Find all subgroups of the octic group D4.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.
- 3. Consider the group under addition. List all the elements of the subgroup, and state its order.Exercises 21. Find all the distinct cyclic subgroups of the octic group in Exercise . 20. Construct a multiplication table for the octic group described in Example of this section.Exercises 19. Find cyclic subgroups of that have three different orders.
- 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 .a. Find all Sylow 3-subgroups of the alternating group A4. b. Find all Sylow 2-subgroups of A4.Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?