Let G act by conjugation on G. (a) Fix g ∈ G. Describe, as precisely as possible, the stabilizer of g under the action of conjugation. (b) Describe, as precisely as possible, the kernel of the action of conjugation.

Q: The kernel of a homomorphism must be normal. Select one: O True O False

A: The kernel of a homomorphism is a normal subgroup.

Q: Show that every nonzero element of Zn is a unit or a zero-divisor

A: The elements of Zn are 0, 1, 2, …, n-1. The non zero elements of Zn are 1, 2, …, n-1. We know that…

Q: a The mapping $ :C → M2 (R) given by (a + bi) = -b is a not one-to-one a ring homomorphism not a…

A: To find - The mapping ϕ : ℂ → M2ℝ given by ϕa + ib = ab-ba is ?     Definition used - A function is…

Q: a. Let Rbe the relation defined on Z by a Rb if a + b is even. Show that R is an equivalence…

A: Note: To show R is a equivalence relation we must show R is reflexive , symmetric and transitive .…

Q: 1. Let V be the set of all ordered pairs of real numbers, and consider the following addition and…

A: (a) u + v =(-1,2)+(3, 4) = (-1+3, 2+4)= (2, 6)ku = 3(-1,2) = (0,3 x 2) = (0,6) (b) The result of the…

Q: Define g(x) to have a domain of {1, 2, 3} and a codomain of N. Which of the following properties…

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Q: U(5) is not isomorphic to U(10) اختر احدى الدجابات True False

A: U(5)=1, 2, 3, 4U(10)=1, 3, 7, 9

Q: Let A=[n].  What are the maximium and minimum possible size of an equivalence relation on A?  Prove…

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Q: R) Prove that there existo a bijective function between the set of all integer numbers Z and the set…

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Q: Let O denote the collection of all open subsets of the real line R. Prove that there is a smallest…

A: Given that Ο denote the collection of all open subset of the real line ℝ. Claim: Borel sigma algebra…

Q: Give an example of two equivalence relations R and S on

A: Consider that R and S are equivalence relations on A=1, 2, 3.

Q: Arithmetical division (÷) is a binary relation on: * O z O Z+ O z- O The set of real numbers

A: The given arithmetic operation is ÷An operation * is said to be binary operation of set S, if…

Q: A. Let R be a relation on set A. A={x : x ∊ J and 0 ≤ x ≤ 12}, and R={(a,b) : |a – b| is multiple of…

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Q: Determine whether the following statements are True or False: - Part A: Suppose R1 and R2 are…

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Q: Find all possible automorphisms of Q(/2).

A: We will use basic knowledge of Galois theory and group theory to solve this question.   Note that an…

Q: Is it possible to define a finite body? If so, give an example and check the property of the…

A: Yes, it is possible to define a finite body.    A finite body is a finite set which is a body,…

Q: Let a be a nonzero algebraic element over F of degree n. Show thata-1 is also algebraic over F of…

A: Given Data The nonzero algebraic element is a≠0.   The nonzero algebraic element a over the F of…

Q: Let A and B be two rings and π:A➔B a ring isomorphism. Prove that "x∈A is a zero divisor if and only…

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Q: A. Let R be a relation on set A. A={x : x ∊ J and 0 ≤ x ≤ 12},  and R={(a,b) :  a - b  is multiple…

A: Since not particular question asked ,as per guidelines solution to only first question (i.e A) is…

Q: Prove that every automorphism of R*, the group of nonzero realnumbers under multiplication, maps…

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Q: Let f: A --> B and g: B --> C. Suppose that g of f is bijective. Show that g is onto and that f is…

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Q: Prove Theorem 2.30 Consider the rule for multiplication in Z„ given by [a][b] = [ab] a.)…

A: PART A- Given: The rule for multiplication in ℤn given by ab=ab We want to show that multiplication…

Q: Construct a nontrivial homomorphism $: Zg → Zı5 >

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Q: If N and K are submodules of M. (i) The kernel of the natural homomorphism N--> M/K is (N+K)/K. (ii)…

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Q: Show that for any k-algebras A1, A2,..., A,, we have (A1 x A2 x ...x A, )P S A" x A" x as algebras.

A: Let, A1,+,*,A2,+,*,...,Ar,+,* be the K-algebras, where + is the addition from 1 to r o which A1 is…

Q: . Prove that the shift o is a homeomorphism.

A: To, prove that σ is homomorphism we will show that the inverse image of open set is open. So, the…

Q: Determine A-" if A is invertible. 1 0 -2 4 -2 1 1 2 -10

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Q: Show that any automorphism of GF( pn) acts as the identity onGF( p).

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Q: *** Let R be a unique factorization domain. Let p E R. If x is irreducible, then it is prime.***

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Q: Arithmetical division (+) is binary relation on Z- Z+ Q O O O O

A: Consider the given information. The binary operation is defined as, If a belongs to G, and b belongs…

Q: let G be the set of all triples of the form (k1, k2,1) or (k1,k2,-1), where the ki, i=1,2 are…

A: Given :  G = k1, k2, ±1 : k1, k2 ∈ℤ with the operation  k1, k2, 1l1, l2, -1 = k1+l2, k2+l1, -1 To…

Q: Prove that is G1 is isomorphic to G2 and H1 is isomorphic to H2, then G1 x H1 is isomorphic to G2 x…

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Q: Let K = ℚ(sqrt2, i ). Find all isomorphisms of K onto a subfield ℚ(bar) as extensions of the…

A: Let K=Q2,i Let ϕ:K→K, So ϕ depends on Q2 and Qi. Now, minimal polynomial of 2 is x2−2 So, other…

Q: 1. Let A, B be any sets with power sets P(A) and P(B), let AC P(A) and BC P(B), and let f : A →→ B.…

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Q: Prove: [a] is an element of Zn has a multiplicative inverse if and only if gcd(a,n)=

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Q: t R be the relation on the set of ordered pairs of positive integers ch that ((a,b), (c,d)) e R if…

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Q: Let a(T), b(T) be polynomials. Show that there exists a unique monic polynomial g(T) such that •…

A: We take an example to satisfy this question

Q: If A is finite, then a) there is an injection of A with a proper subset of itself. b) there is a…

A: Given that set A is a finite set. Let B be a proper subset of finite set A. We check each option by…

Q: 5. Let K = Q(V2, i). Find all isomorphisms of K onto a subfield Q as extensions of the identity…

A: (5) Given, K=Q2, i Let φ:K→K,  So, φ depends on φ2 and φi Now, minimal polynomial of 2 is x2-2, so,…

Q: Produce an example of a subset of 1R³ that is Closed under addition, but not closed under scalar…

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Q: 5. Let K = Q(V2, i). Find all isomorphisms of K onto a subfield Q as extensions of the identity…

A: Hello. Since your question has multiple parts, we will solve the first  part for you. If you want…

Q: Show that for distinct i and j from {1,...n} there are exactly two auto- morphisms of Cn which send…

A: i and j from {1,...n} there are exactly two auto- morphisms of Cn which send i to j. Then Show that…

Q: Let G= Find all the conjugate classes of (G,.). (i.e G={1,a,b,a.b}).

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Q: Give a binary relation R on an n-element set such that R != ∅ and |R ◦ R^(−1) | = n^(2) · |R^(−1) ◦…

A: Given:  R != ∅ and |R ◦ R^(−1) | = n^(2) · |R^(−1) ◦ R|

Q: Let B1 and B2 be two σ-algebras on a set X. Define B = B1 ∩ B2 by A ∈ B if and only if A ∈ B1 and A…

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Q: (a) Prove that if , and 2 are segma algebras over X, then nL, is a sigma algebra over X. (b) Is ),…

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Q: Let R be a relation on the set of all integers such that a Rb if and only if a +b = 0. (a) Is R…

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Q: Q2: State and prove the first isomorphism theorem.

A: Let be a group map, and let be the quotient map. There is an isomorphism such that the…

Q: Let F = {{n} : n ∈ Z} and G = {{r} : r ∈ R}. Find the smallest σ-algebra in Z containing F and the…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.