1/ Determine in each of the parts if the given mapping is a homomorphism. If so, identify its kernel and whether or not the mapping is 1-1 or onto. (a) G = Z under +, G' = Zn, 4(a) = [a] for a € Z. 1 (b) G group, : G→ G defined by 4 (a) = a¹ for a € G. (c) G abelian group, 4: G→ G defined by (a) = a¹ for a E G.

Q: 4. Suppose (G₁, *) and (G₂, 0) are groups, let i and J denote the respective identities of G₁ and…

A: and are group. be a group isomorphism.We have to prove that if  has finite order the  has finite…

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A: Since you have posted a question with multiple subparts, we provide the solution only to the first…

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A: see below the answer

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Q: 4. (a) Show that every group of order 4 is isomorphic to either Z4 or V4. (b) Show that H {1,…

A: We have show given property:

Q: 3. Let G, H, K be finitely generated Abelian groups. (i) Show that if G G HH then G= H. (ii) Show…

A: We will solve this by the help of isomorphism

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A: In this question, we have to determine the Galois group of x3 - 1 over Q and x3 - 2 over Q.

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Q: 17*. Find an explicit epimorphism from A5 onto a group of order 3

A: Epimorphism: A homomorphism which is surjective is called Epimorphism.

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A: Given: "G" is an abelian group,  We need to prove that a×bn=an×bn for all integers n. Abelian Group…

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A: Lets solve the question.

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Q: Determine the class equation for non-Abelian groups of orders 39and 55.

A: We have to determine the class equation for non-Abelian groups of orders 39 and 55.

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Q: Let G be a group with |G| = 96. Suppose H and K are subgroups of G with H| = 12 and K| = 16. Then…

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Q: 2* Let f G H be a group homomorphism. Prove that if x E G and n is a natural number then f(x)= f(x)"

A: To prove the required property of group homomorphisms

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Q: 15*. Find an explicit epimorphism from Z24 onto a group of order 6. (In your work, identify the…

A: To construct a homomorphism from Z24 , which is onto a group of order 6.

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Q: Consider the following map defined on the symmetric group S3: Þ: S3 → GL₂ (R) given by: Þ(e) =…

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Q: Suppose now that we have two groups (X,o) and (Y, *). We are familiar with the Cartesian product X x…

A: Let X,◊ and Y,* are two groups. The Cartesian product of X and Y defined by X×Y=x,y | x∈X and  y∈Y.…