a) Check whether G = (3n, ·) is a group, where n ∈ Z. b) Let p1 (x) = x3 + 1 and p2 (x) = x4 + x. Calculate p1 (x) + p2 (x) and p1 (x) · p2 (x) in Z2 [x] x3 + x. (c) Find the set Z3 [x] x2 + 1

Q: Let 3 denote the centre of a group G. then show that a 3 iff N(a) = G. also show that if G is…

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Q: 2. Let G = (R*, .), the multiplicative group of nonzero real numbers. Let H = {x ER: x -1} be the…

A: 2. Let , the multiplicative group of non-zero real numbers.Let  be the group with operation .Prove…

Q: Go Let G be a group and a, b = G. Prove that (ab)−¹ = b¯¹a-¹. Let G be a group. Prove that for all…

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Q: Give an example of a group G and elements x, y E G with o(xy) o(x)o(y).

A: (a) Give an example of a group G and elements x, y∈G with oxy<minox, oy. Let us take the group…

Q: Let C(x) represent the centralizes of a. If x is an element of group G with the order of x be 5 (|x|…

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Q: Let G and H be groups and consider the function f: G → G × H defined by f(g) = (g, eH ) where eH is…

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Q: Let G be the set consisting of the following matrices: (6 9) (6 ) ( ) -1/2 v3/2 V3/2 1/2 -1/2 V3/2)…

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Q: (4) Let (G, *) be a group and assume that z, a, b € G. Let also that c = x *a*r¹ and d = x * b*x-¹.…

A: Let  be  a  group  and  assume  that Suppose  that Since,  and  is  a  group, it follows  that…

Q: let x and g be elements of a group G. show that x and gxg-1 have the same order. now prove that xy…

A: Hence, proved.Explanation:Step 1:To prove that x and gxg-1  have the same order for any elements x…

Q: Let G be a group. Assume a and b are elements of G and set H₁ = and H₂ =. Assume |H₁| = 21 and |H₂|…

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Q: (c) If K is a subgroup of G, then p(K) is a subgroup of H. Given: K < G (d) If K' is a subgroup of…

A: I have mentioned the test used to prove a subset to be a subgroup, as a note above. You can skip it…

Q: f G is a finite group and Z is its centre, then class equation of G is expressible as D(6) 0(N(a))…

A: G is a finite group and Z is its centre, then we have to show that the  class equation of G is…

Q: (B) Let G be a set of all real numbers except 0. Define on G by a b = |a|b, where lal is the…

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Q: Let G and G' be groups. (a) If p: G → G' is a function, what is required for p to be a homomorphism?…

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Q: For a group G and a fixed element a ∈ G, define the subset C(a) to be the set of all elements in G…

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Q: To which group is Z12/ ker f isomorphic to? (a) Z₁ (b) V4 (c) D4 (d) A₁

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Q: Let G, H, and K be groups. Show that (G x H) × K = G × (H × K). Hint: An element of (G × H) × K will…

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Q: 5 Let G be the set of all functions f: R → R of the form f(x) = ax + b where a and b are real…

A: Introduction: Function composition is an operation where two functions are combined to form a new…

Q: Demonstrate or explain why the system (P2(R#), *) is NOT a semi-group, that is, demonstrate or…

A: To demonstrate or explain why: P2ℝ,* is NOT closed or P2ℝ,* is NOT associative

Q: Find a matrix M such that M-¹0 (8) M = (19) for every element g € S3. Conclude that the…

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Q: Consider the group D4

A: Given: Group D4=a,b=e=1,a,a2,a3,b,ab,a2b,a3b and a=1 2 3 4, b= 2, 4 To find : The value of…

Q: Find an isomorphism o from the multiplicative group G of nonzero complex numbers to the…

A: We need to find ϕ for the multiplicative group G of non zero complex numbers to the multiplicative…

Q: Explain why the group (Z4, +) is NOT isomorphic to the group (LI, ∘) Group (LI ∘): The symmetries…

A: If two groups are isomorphic then they should have same number of elements

Q: 4. Which of the groups U(14), Z6, S3 are isomorphic?

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Q: Determine the galois group

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Q: Show that U(8) is the internal Sivect product of certain two groups of U(8)

A: To show that U(8) is the internal direct product of certain two subgroups of U(8), we need to find…

Q: f G is a finite group and Z is its centre, then class equation of G is expressi o(G) o(G) = o(Z) + E…

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Q: Let G1 and G2 be groups. Prove that G1 × G2 is isomorphic to G2 × G1.

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Q: (d) Find the cosets of the quotient group (5)/(10), and determine its order.

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