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Chapter 3 Solutions
Elements Of Modern Algebra
- 32. Let be a fixed element of the group . According to Exercise 20 of section 3.5, the mapping defined by is an automorphism of . Each of these automorphism is called an inner automorphism of . Prove that the set forms a normal subgroup of the group of all automorphism of . Exercise 20 of Section 3.5 20. For each in the group , define a mapping by . Prove that is an automorphism of .arrow_forward5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:arrow_forwardLet A be a given nonempty set. As noted in Example 2 of section 3.1, S(A) is a group with respect to mapping composition. For a fixed element a In A, let Ha denote the set of all fS(A) such that f(a)=a.Prove that Ha is a subgroup of S(A). From Example 2 of section 3.1: Set A is a one to one mapping from A onto A and S(A) denotes the set of all permutations on A. S(A) is closed with respect to binary operation of mapping composition. The identity mapping I(A) in S(A), fIA=f=IAf for all fS(A), and also that each fS(A) has an inverse in S(A). Thus we conclude that S(A) is a group with respect to composition of mapping.arrow_forward
- Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.arrow_forwardExercises 22. Let be a finite cyclic group of order with generators and . Prove that the mapping is an automorphism of .arrow_forwardProve or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.arrow_forward
- 13. Assume that are subgroups of the abelian group . Prove that if and only if is generated byarrow_forwardLet G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.arrow_forward23. Let be a group that has even order. Prove that there exists at least one element such that and . (Sec. ) Sec. 4.4, #30: 30. Let be an abelian group of order , where is odd. Use Lagrange’s Theorem to prove that contains exactly one element of order .arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,