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 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 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 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
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.8: Some Results On Finite Abelian Groups (optional)
Problem 14E: Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic...
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Question 1to3
![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%2F1c7265f2-9dda-4402-bf47-87522a081370%2Fd3e35e68-55c3-4bc8-888e-52ce0727f996%2Fcviaxwd_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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