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
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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Question
the first 3 question
![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[√](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f3bb0f4-e265-4285-b651-d4e92566d76d%2F2035ae53-4742-4b40-b2fc-2e5c4772a854%2F45vll0p_processed.jpeg&w=3840&q=75)
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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