If f (x) is a cubic irreducible polynomial over Z3, prove that either xor 2x is a generator for the cyclic group (Z3[x]/<f (x)>)*.

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A: image is attached

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A: *By Bartleby policy I have to solve only first one as these are all unrelated and very lengthy…

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Q: Question 1. Suppose that G = xy = yx². {e, x, x², y, yx, yx²} is a non-Abelian group with |æ| = 3…

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A: Detailed explanation mentioned below

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Q: Consider the group G = {x € R such that x # 0} under the binary operation *: x*y=-2xy O x*x*x=4x^3 O…

A: Thanks for the question :)And your upvote will be really appreciable ;)

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Q: Q3: (A) Prove that 1. There is no simple group of order 200.

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A: We will use property of homomorphism to solve the following question

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Q: Let a € Zz be a zero of the irreducible polynomial p(x) = x3 + x² + 2 in Z3[x]. a. What is |Z3(a)|?…

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Q: Consider the group G = {x € R such that x # 0} under the binary operation x*y=-2xy O x*x*x=-x^3/4 O…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: . Prove that the group Zm × Zn is cyclic and isomorphic to Zmn if and only if (m, n) = 1.

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Q: 22. Prove that the set = {(₁ ~ ) 1} x) | : x, y ≤ R, x² + y² = 1 = SO(2) = forms an abelian group…

A: Given: 22. SO(2)=x-yyx : x, y∈ℝ, x2+y2=1 To show: The given set is a group with respect to…

Q: Consider the group G = {x E R|x # -1/2} under the binary operation*: X * y = 2xy – x + y – 1. The…

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Q: 1. Consider the groups (R+, ) and (R,+). Then R* and R are isomorphic under the mapping $(x) = log10…

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Q: Show if all primitive transformations of the nonzero form x '= x ,y' = cx + dy d are a group.

A: