Let
(This mapping is called the trace of the matrix.) Prove or disprove that
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Elements Of Modern Algebra
- 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.arrow_forwardLet G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.arrow_forward11. Let be , and let be the group of nonzero real numbers under multiplication. Prove that the mapping defined by Is a homomorphism, and find ker . Is an epimorphism? Is a monomorphism? (The value of this mapping is called the determinant of the matrix.)arrow_forward
- Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.arrow_forward39. Let be the set of all matrices in that have the form for arbitrary real numbers , , and . Prove or disprove that is a group with respect to multiplication.arrow_forwardTrue 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.arrow_forward
- Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .arrow_forwardProve or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.arrow_forwardLet A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)arrow_forward
- 9. Suppose that and are subgroups of the abelian group such that . Prove that .arrow_forwardExercises 22. Let be a finite cyclic group of order with generators and . Prove that the mapping is an automorphism of .arrow_forwardLet G be an abelian group. For a fixed positive integer n, let Gn={ aGa=xnforsomexG }. Prove that Gn is a subgroup of G.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,