Let G1 and G2 be groups with respect to addition. Define equality and addition in the Cartesian product by G1G2 (a,b)=(a,b) if and only if a=a and b=a (a,b)+(c,d)=(ac,bd) Where indicates the addition in G1 and indicates the addition in G2. Prove that G1G2 is a group with respect to addition. Prove that G1G2 is abelian if both G1 and G2 are abelian. For notational simplicity, write (a,b)+(c,d)=(a+c,b+d) As long as it is understood that the additions in G1 and G2 may not be the same binary operations. (Sec. 3.4,27, Sec. 3.5,14,15,27,28, Sec. 3.6,12, Sec. 5.1,51) Sec. 3.4,27 Prove or disprove that each of the following groups with addition as defined in Exercises 52 of section 3.1 is cyclic. a. 23 b. 24 Sec. 3.5,14,15,27,28, Consider the additive group of real numbers. Prove or disprove that each of the following mappings : is an automorphism. Equality and addition are defined on in Exercise 52 of section 3.1. a. (x,y)=(y,x) b. (x,y)=(x,y) Consider the additive group of real numbers. Prove or disprove that each of the following mappings : is an isomorphism. a. (x,y)=x b. (x,y)=x+y Consider the additive groups 2, 3, and 6. Prove that 6 is isomorphic to 23. Let G1, G2, H1, and H2 be groups with respect to addition. If G1 is isomorphic to H1 and G2 is isomorphic to H2, prove that G1G2 is isomorphic to H1H2. Sec. 3.6,12 Consider the additive group of real numbers. Let be a mapping from to , where equality and addition are defined in Exercise 52 of Section 3.1. Prove or disprove that each of the following mappings is a homomorphism. If is a homomorphism, find ker , and decide whether is an epimorphism or a monomorphism. a. (x,y)=xy b. (x,y)=2x Sec. 5.1,51 Let R and S be arbitrary rings. In the Cartesian product RS of R and S, define (r,s)=(r,s) if and only if r=r and s=s (r1,s1)+(r2,s2)=(r1+r2,s1+s2), (r1,s1)(r2,s2)=(r1r2,s1s2). a. Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of R and S and is denoted by RS. b. Prove that RS is commutative if both R and S are commutative. c. Prove that RS has a unity element if both R and S have unity elements. d. Give an example of rings R and S such that RS does not have a unity element.
