Problem 1. In each of the following problems, determine if the given is a group homomorphism. 1.1.4: Z/12Z → Z/6Z given by ([k]) = [2k] for all k Є Z. 1.2. Let G and H be any two groups, for all gЄ G and hЄ H. and let : Gx H → G be given by (g, h) = g 1.3. & : GL2(R) → GL2 (R), given by (A) = AT for all A = GL2(R). 1.4. Let G be any abelian group, and let : G → G be given by (g) = g² for all g Є G. Remark. In Problem 1.1. you should explain whether or not is a function at all: is it well-defined? That is, if k and are both representatives of the same class in Z/12Z, do we have ([k]) = ([l])?

1. 2. Are the groups Z/2Z x Z/12Z and Z/4Z x Z/6Z isomorphic? Why or why not?

10.3.3. Let G = Z/4Z × Z/6Z, and let H be the subgroup of G generated by (2,2). (a) What are the elements of H? (b) What are the elements of G/H? (c) Find a familiar group that is isomorphic to G/H.

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.

11.1.12. Let G be a group, and assume that H and K are normal subgroups of G with trivial intersection. Further assume that G = HK. Prove that GH × K.

7. [10] Let G be a group, with x E G and o (x) = 48. Give the order of the following elements of G: a) o(x²) b) o (x20) c) o(x23)

2.3.3 Subgroups

(4.5) #2 Modern Applied Algebra. Show work.

6. Show that for any two elements x, y of any group G, o(xy) = o(yx). %3D

. Let H (27) be the subgroup of G=Z36. What is the order of the quotient group G/H? = (a) 3 (b) 6 (c) 9 (d) 12