Define ∗ on Z by x ∗ y = x + y + 4. Then: (a) Prove that ∗ is a binary operation on Z.  (b) Show that ∗ is an associative operation on Z.  (c) Determine the identity element for ∗ in Z.

Q: Let ø be an onto homomorphism from G1 onto G2. (a) Prove that if G1 is abelian, then G2 is abelian.…

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Q: Let L be a finite dimensional Lie algebra. Any finite dimensional representation of L is completely…

A: Let L be a finite dimensional Lie algebra. From this, we have Every finite dimensional…

Q: ). Let R be binary relation on N defined by xRy if and only if x- 2s ys x+2. Is R reflexive? Is R…

A: We will usethe definition of reflexive, symmetric, antisymmetric and transitive realtion to answer…

Q: invertible

A: Given that: Let R be a ring with one and a, b two elements in R so that ab = ba Let (a-b) ^ n = 0 ,…

Q: Let ∗ be the binary operation on Z defined by x∗y=x+xy. Then (Z,∗) has a right identity element but…

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Q: Q2. (a) Prove that if , and >. are segma algebras over X, then >n), is a sigma algebra over X. (b)…

A: A Collection ∑1 of subsets of an arbitrary space X is said to be a sigma algebra if:(i) X∈ ∑1(ii)…

Q: Q2. (a) Prove that if , and E, are segma algebras over X, then . Σ is a sigma algebra over X. (b) Is…

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Q: G be defined by f(r) = x1. Prove that f is operation-preserving if 6*. Let G be a group and f: G and…

A: To prove that the given function f is a homomorphism (operation preserving) if and only if G is…

Q: Let G = {[1 0 0 1] ,[−1 0 0 1] ,[1 0 0 −1], [−1 0 0 −1]} . Is G ∼= K4? If yes, give an explicit…

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Q: integer 16. If EG→His a surjective homomorphism of groups and G is abelian, prove that H is abelian.…

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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: Determine all homomorphisms from Z4 to Z2 ⨁ Z2.

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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 Φ be an automorphism of C*, the group of nonzero complexnumbers under multiplcation. Determine…

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Q: Suppose that A⊆G generates G. Prove that G is abelian if and only if any two elements of A commute…

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Q: 3. Are the rings 2Z, 3Z isomorphic. Any isomorphism is linear for addi- tion, so it has the form f…

A: Ring Isomorphism: Letus consider two rings R and S.Let us consider the mapping  f: R→S. Then it is…

Q: Show that the function ø : ℝ → ℝ defined by ø (x) = x3 is an isomorphism of the binary structure ⟨ ℝ…

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Q: Prove that if Σ1 and Σ2 are sigma algebras over X, then Σ1 ∩ Σ2 is a sigma algebra over X.

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Q: Prove or disprove that union of two sigma algebra is a sigma algebra

A: See the attachment.

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

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Q: Show that there is no associative algebra A such that the Lie algebra arising from A is Glo(C)

A: The center of a lie algebra L is, C(L)={x∈L:[x,y]=0 for all y∈L} Since[x,y]=0 so x and y commutes.…

Q: Compute ø(23) for the homomorphism ø:Z→Z, such that ɖ1) =5

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Q: In the set of real numbers R there is an operation defined as: x x y = Vx³+y³ prove that (R, x) is…

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Q: Q2. (a) Prove that if , and E, are segma algebras over X, then EnE, is a sigma algebra over X. (b)…

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Q: Define a binary operation on R\ {0}by x*y = . 2 Prove that this set with this binary operation is an…

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

A: we have φ:z2×z3→z3×z2 we know a result: A G1⊕G2 is isomorphic to G2⊕G1 by g1,g2→g2,g1.

Q: Q2. (a) Prove that if ), and >. are segma algebras over X, then 2n2, is a sigma algebra over X. (b)…

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Q: (2) Let R be binary relation on N defined by rRy if and only if r <y< 2r. Is R reflexive? Is R…

A: We have given that R be a binary relation on ℕdefined by xRy if and only if x≤y≤2x. Now we have to…

Q: Compute the Killing form for the Heisenberg algebra and find its radical.

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Q: Q. If >, and E, are segma algebras over X, 12 Eu, is a sigma algebra over X а. '2 b. ,n, is a sigma…

A: Correct answer is option B Explanation is given below

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

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Q: Q. If E, and E, a. E,n E, is a sigma algebra over X b. EuE, is a sigma algebra over X E, E, is a…

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Q: Recall that Q8 =. Prove that the map o from Qg to GL2(C) defined on generators by P(i) = |V-1 And…

A: A group homomorphism Φ : G→H is a map such that for any g1, g2 ∈G, Φg1g2=Φg1Φg2. The kernel of a…

Q: Suppose that f:G-G such that f(x)- axa. Then fis a group homomorphism if and only if O a*2e O an4e O…

A: Here we will evaluate the required condition.

Q: a) The neutral element 0a of an A ring is unique. b) The opposite -a of an element a, of an A ring,…

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Q: 1. Given the binary relation R on R defined by „R, if and only if x > y – 1, for each of the…

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Q: Q/ Let * defined on the set of integers by a*b = b, one of the following is false : 1) * is binary…

A: a*b=b

Q: 6. Describe all extensions of the automorphism ý 3.-/3 of Q(v3) to an isomorphism mapping Q(i, 3, 2)…

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Q: Define bi V3 and bn+1 3bn. Show that {bn} is monotone and compute lim,n- bn.

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Q: Prove that An is not abelian for all n > 4.

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Q: Q. If E, and , are segma algebras over X, EuE, is a sigma algebra over X b. E E.E, is a sigma…

A: Given that, correct answer is option B Explanation is given below

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

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Q: → R be a ring homomorphism, where R is a commutat .. bn be some arbitary elements of R. then there…

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Q: In the set of real numbers IR there is an operation defined as: I x y= Vr³+y³ %3D prove that (IR, ×)…

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Q: Σ Q. If E, and E, E,n£, is a sigma algebra over X b. E,uE, is a sigma algebra over X .E,\E, is a…

A: The solution is given by using the definition of sigma algebra as follows