3. Let G = Z4xZ7. a. Determine all the generators for G. b. Determine a subgroup for each order 14, 7, 4, and 2. Then draw a subgroup lattice diagram of G. c. Let N = ((0, 1)). Determine the right cosets of N.

Find two groups of order 6 that are not isomorphic.

Exercises 21. Suppose is a cyclic group of order. Determine the number of generators of for each value of and list all the distinct generators of . a. b. c. d. e. f.

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.

Find the right regular representation of G as defined Exercise 11 for each of the following groups. a. G={ 1,i,1,i } from Example 1. b. The octic group D4={ e,,2,3,,,, }.

Exercises 10. For each of the following values of, find all subgroups of the cyclic group under addition and state their order. a. b. c. d. e. f.

Let G= a be a cyclic group of order 35. List all elements having each of the following orders in G. a. 2 b. 5 c. 7 d. 10

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 ] .

9. Determine which of the Sylow p-groups in each part Exercise 3 are normal. Exercise 3 3. a. Find all Sylow 3-subgroups of the alternating group . b. Find all Sylow 2-subgroups of .

12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.