Define the mapping 7: R²→R by π((x,y))=x. (Note that R is a group under addition with identity 0). Prove that is a homomorphism. Find the kernel of T. a) b)
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- 14. Let be an abelian group of order where and are relatively prime. If and , prove that .31. (See Exercise 30.) Prove that if and are primes and is a nonabelian group of order , then the center of is the trivial subgroup . Exercise 30: 30. Let be a group with center . Prove that if is cyclic, then is abelian.45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )
- Exercises 22. Let be a finite cyclic group of order with generators and . Prove that the mapping is an automorphism of .14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .
- True or False Label each of the following statements as either true or false. 11. The invertible elements of form an abelian group with respect to matrix multiplication.let Un be the group of units as described in Exercise16. Prove that [ a ]Un if and only if a and n are relatively prime. Exercise16 For an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have multiplicative inverses. Prove that Un is a group with respect to multiplication.Let H be a torsion subgroup of an abelian group G. That is, H is the set of all elements of finite order in G. Prove that H is normal in G.