Problem 5. Let G be a group and H ≤ G be a subgroup. Define a relation ~ on G by xy if y¹× Є H. 5.1. Show that ~ is an equivalence relation on G with [x] = xH = {xh : hЄH} \ xЄ G. 5.2. Show that for each x EG, the function fx : H → [x] given by fx(h) all hЄ H is bijective. Is it a group homomorphism? = xh for 5.3. Use your solutions to Problems 5.1 and 5.2 to show that if G is a finite group, then |H| divides |G|.

5. Let G be a group, let f, g = G, and suppose that f~G 9. Show that the order of f is equal to the order of g.

2.13. Let G and H be groups. A function : G→ H is called a (group) homomor- phism if it satisfies (91 * 92) = (91) * (92) for all 91,92 € G. (Note that the product g₁ ★ 92 uses the group law in the group G, while the prod- uct (91) * (92) uses the group law in the group H.) 108 Exercises (a) Let eg be the identity element of G, let e be the identity element of H, and let g E G. Prove that o(еG) = еH and o(g¯¹) = þ(g)−¹. (b) Let G be a commutative group. Prove that the map : G → G defined by (g) = g² is a homomorphism. Give an example of a noncommutative group for which this map is not a homomorphism. (c) Same question as (b) for the map ø(g) = g¯¹.

Draw a directed graph diagram for relation S on {0, 1, 2, 3,4}, where TSy iff x + y = 4