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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. -1 (3). and H₂ in G are finite and coprime to each other. Then 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 9₁ and 92 both acts transitively on {1, 2,..., 100 }, then there exists an element h E S100 such that hg₁h¹ = 92. -1 (3). and H₂ in G are finite and coprime to each other. Then Let H₁ and H₂ be two normal subgroups of G. Suppose the indexes of H₁

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 hg₁h-¹ = 92. (3). and H₂ in G are finite and coprime to each other. Then Let H₁ and H₂ be two normal subgroups of G. Suppose the indexes of H₁ G/H₁ H₂ (G/H₁) × (G/H₂). ≈ 2
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 hg₁h-¹ = 92. (3). and H₂ in G are finite and coprime to each other. Then Let H₁ and H₂ be two normal subgroups of G. Suppose the indexes of H₁ G/H₁ H₂ (G/H₁) × (G/H₂). ≈ 2
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