2. Let G and H be any two groups. Define the set G x {e} = {(g, eμ) | gЄG} which is a subset of G × H. a) Show that the inverse of (a, b) E G x H is (a¹, b¹). b) Prove that G x {e} is a normal subgroup of G × H. c) Show that f: G x H→ H defined by f((g, h)) = h is a surjective homomorphism. d) Find the kernel of f. e) Apply the First Isomorphism Theorem (of groups) to the function 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,,,, }.

18. If is a subgroup of , and is a normal subgroup of , prove that .

For each a in the group G, define a mapping ta:GG by ta(x)=axa1. Prove that ta is an automorphism of G. Sec. 4.6,32 Let a be a fixed element of the group G. According to Exercise 20 of Section 3.5, the mapping ta:GG defined by ta(x)=axa1 is an automorphism of G. Each of these automorphisms ta is called an inner automorphism of G. Prove that the set Inn(G)=taaG forms a normal subgroup of the group of all automorphisms of G.

24. The center of a group is defined as Prove that is a normal subgroup of .

Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1 isomorphic to G2/H2. Determine the possible orders of H1 and H2 under the following conditions. a. G1=24 and G2=18 b. G1=32 and G2=40

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

18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.

16. Let be a subgroup of and assume that every left coset of in is equal to a right coset of in . Prove that is a normal subgroup of .

Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.