Exercise 3.3. Let G be a group. Show that a subset H is a subgroup of G if and only if H is nonempty and ab¹ € H for all a, b = H.

9. Find all homomorphic images of the octic group.

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.

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?

Let be a group of order 24. If is a subgroup of , what are all the possible orders of ?

If H and K are arbitrary subgroups of G, prove that HK=KH if and only if HK is a subgroup of G.

Find subgroups H and K of the group S(A) in example 3 of section 3.1 such that HK is not a subgroup of S(A). From Example 3 of section 3.1: A=1,2,3 and S(A) is a set of all permutations defined on A.

Let be a subgroup of a group with . Prove that if and only if .

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

5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19: