Find a rule for the following isomorphism: 9: Z2 x Z3 + Zz x Z2 p (a, b) = (

Q: Let G=Z5xZ6X Z9 X Z12- Find the elementary divisors and invariant factors of G.

A: Solution :-

Q: How would you use the first isomorphic theorem to prove that G/K is isomorphic to G?

A: The objective of the question is to use the first isomorphism theorem to prove that the quotient…

Q: Explain why the correspondence x → 3x from Z12 to Z10 is not ahomomorphism.

A: We have to explain that: Why the correspondence x→3x from Z12 to Z10 is not a homomorphism.

Q: Prove that G1 ⨁ G2 is isomorphic to G2 ⨁ G1. State the general case

A: To Prove: G1⊕G2 is isomorphic to G2⊕G1. Also, state the general case.

Q: Find Aut(Z6).  Define each automorphism and justify your conclusion.

A: Concept:

Q: (c) Give one example where the Green theorem fails, and explain how.

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Q: Suppose that x and y are real numbers such that y is 6 greater than x. What is the smallest possible…

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Q: How would you find an isomorphism from f: Z3 X Z14 -> Z21 X Z2. Kindly elborate.

A: Given That : A mapping f :Z3×Z14→Z21×Z2 To Find : The isomorphisms from f :Z3×Z14→Z21×Z2.

Q: 2. Is the following set of polynomials linearly independent in P2? Explain your work. (-1+2t + 3t?),…

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Q: Does the following set of polynomials {t? + 1, t2 + t,t + 1} span P2?

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Q: A PSR is the congruent image of A PQR after a transformation. P R True False

A: Given: The statement, "△PSR is the congruent image of △PQR after a transformation" in figure , To…

Q: Is U (8) ∼= Z2 ×Z2 ? why? If yes, gives an explicit isomorphism.

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Q: Determine all homomorphisms from D5 onto Z₂ Z₂. Determine all homomorphisms from D5 to Z₂ + Z₂.

A: We have to determine all homomorphism from  onto .We also have to find homomorphism from  to

Q: Let a, b, and c be distinct real numbers. Show that…

A: Consider the equation: We have to check whether it is a zero polynomial, symmetric and bi-linear or…

Q: (a) Consider the set A = {x² E Q1x ER}. Determine whether A is closed under addition and…

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Q: Let T: R² R² be given by T(x1, x₂) = -[at] It is known that I is an isomorphism. T-¹ is given by the…

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Q: express Z10 as a product of Z5 x Z2 , verify that both groups are isomorphic.

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Q: Find gcd (272, 1479) and find the integers x, y such that gcd(272, 1479) = 272x + 1479y

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Q: The Cartesian product A × A has 9 elements among which are found (–1, 0) and (0,1). Find the set A…

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Q: Show thet tegular a S(G) = A(G). G

A: Let us assume that graph G is regular. Now,  δ(G)= degree of the vertex with minimum degree…

Q: .Let Z2 be an algebraic closure of Z2, and let a, ß e Z2 be zeros of x3 x2+1 and of x3 +x + 1,…

A: We know that, a function h(x) (say, of degree 3) is reducible over ℤ2, if and only if it has zero in…

Q: 2. Let A, B, C, and D be four groups such that A-C and B D. Show that A x B Cx D.

A: We will use the face that A isomorphic to C  and B isomorphic to D to prove the asked result.

Q: D A B D Zax Zo Z₂ Z₂ x Z₂ ({0},+) E Z₁ Which one of the following is isomorphic to Z4 x Z6/((1,1))?

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Q: Let GZ16 × Z4. (i) Construct a surjective homomorphism : G→ Z8. (ii) What is ker()? (iii) Show that…

A: Given: G=ℤ16×ℤ4   (i.) To construct: A surjective homomorphism φ:G→ℤ8. (ii.) To determine: kerφ…

Q: G be H and Ki Then direct 7(H) and Let direct pradeuct 7(G) Yoduct z'(k). is

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Q: Define the cartesian product X × Y of two sets X and Y, and the power set P(X) of X. Find X x Y and…

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