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.

Problem 1 : G is a group. Define a relation ∼ on G by a ∼ b if a = b or a = b^(−1) . (1) Show that this relation is an equivalence relation. Describe the equivalence classes. (2) Prove that |G| is an odd number if and only if the number of elements of order 2 is even. (3) If |G| is an even number, G must contain a subgroup of order 2