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Determine whether the following statements are true or false. Justify your answer with brief explanations or counterexamples. (1). An abelian group of order 81 is a cyclic group. (2). Suppose 9₁ and 92 are two cycles in S100. If the length of cycles g₁ and 92 both acts transitively on {1, 2, , 100 }, then there exists an element h E S100 such that hgih-1 = 92. (3). and H₂ in G are finite and coprime to each other. Then G/H₁ H₂ (G/H₁) × (G/H₂). Let H₁ and H₂ be two normal subgroups of G. Suppose the indexes of H₁

Determine whether the following statements are true or false. Justify your answer with brief explanations or counterexamples. (1). An abelian group of order 81 is a cyclic group. (2). Suppose 9₁ and 92 are two cycles in S100. If the length of cycles g₁ and 92 both acts transitively on {1, 2, , 100 }, then there exists an element h E S100 such that hgih-1 = 92. (3). and H₂ in G are finite and coprime to each other. Then G/H₁ H₂ (G/H₁) × (G/H₂). Let H₁ and H₂ be two normal subgroups of G. Suppose the indexes of H₁

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.7: Direct Sums (optional)
Problem 16E
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Determine whether the following statements are true or false. Justify your answer with brief explanations or counterexamples. (1). An abelian group of order 81 is a cyclic group. (2). Suppose 9₁ and 92 are two cycles in S100. If the length of cycles 9₁ and 92 both acts transitively on { 1, 2, ..., 100 }, then there exists an element h E S100 such that hg₁h¹ = 92. (3). Let H₁ and H₂ be two normal subgroups of G. Suppose the indexes of H₁ and H₂ in G are finite and coprime to each other. Then (4). (5). mial ring C[x]. (6). G/H₁ H₂ (G/H₁) × (G/H₂). 2 The ring F[x]/(x²) has a unique maximal ideal. The set {f € C[x] | ƒ(2) = 0, ƒ (0) = 0} is a principal ideal of the polyno- The field Q[√5] is the field of quotients of Z[√
Transcribed Image Text:Determine whether the following statements are true or false. Justify your answer with brief explanations or counterexamples. (1). An abelian group of order 81 is a cyclic group. (2). Suppose 9₁ and 92 are two cycles in S100. If the length of cycles 9₁ and 92 both acts transitively on { 1, 2, ..., 100 }, then there exists an element h E S100 such that hg₁h¹ = 92. (3). Let H₁ and H₂ be two normal subgroups of G. Suppose the indexes of H₁ and H₂ in G are finite and coprime to each other. Then (4). (5). mial ring C[x]. (6). G/H₁ H₂ (G/H₁) × (G/H₂). 2 The ring F[x]/(x²) has a unique maximal ideal. The set {f € C[x] | ƒ(2) = 0, ƒ (0) = 0} is a principal ideal of the polyno- The field Q[√5] is the field of quotients of Z[√
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