1) a) Are G1 and G2 Isomorphic? Why? b) Are A and B Isomorphic? Why? A

Q: Show that U(8) and U(10) are not isomorphic.

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Q: Prove that the dual of (R², || · ||,) is isometrically isomorphic to (R², || · .).

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Q: Let 2 = {a, b, c}. Write down all possible o-algebras on N.

A: Given: Ω=a,b,c

Q: 10 2. Use De Moivre's theorem to determine 1 in the form a + ib. 2

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Q: Prove that Z is countable, exhibiting an explicit bijection between N and Z.

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Q: 3. If x is uncountable, then prove that A = {ECX: ECX: algebra O X: E is countable or E² is…

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Q: 5. Prove OR disprove that Z/2ZOZ/3Z is isomorphic to Z/6Z

A: To prove or disprove that  Recall that , Now to prove or disprove that  These elements of these…

Q: 5- The number of ring isomorphism from the ring of integers (Z+.) onto itself is: (a) Zero (b) one…

A: Option (b) is correct.

Q: Find a bosis for the nullspace of the morkit 5-23 (-12 -2-10 4 -6J

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Q: Prove that R under addition is not isomorphic to R* under multiplication.

A: we know that R and R* = R – {0} are groups under addition and multiplication respectively. We have…

Q: Prove that if x is even and y is even, then the product of x and y is divisible by 4.

A: We use the Division Algorithm to prove the given statement.

Q: Show that there exist only two homomorphism from R into R?

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Q: Prove that if gcd (a,b)=1>then gcd (ab,ab)=l

A: I have proved using the fact that gcd can be expressed as integers linear combination q

Q: Z6 is isomorphic to U(7) by giving an isomorphism φ from Z6 to U(7).

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Q: PROVE |z.|-|z,|<|z, – z||

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Q: Exercise 3.2.3 Show that Z, under addition is isomorphic to Z; under multiplication.

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Q: (b) Show that 1+ 2/2 has infinite order in R*.

A: Given that R=m+n2 | m,n∈ℤ (b) The objective is to show that 1+22 has infinite order in ℝ×

Q: 4 3. Let a, b, c, d E R. Define explicitly a bijection from [a, b] [c, d]. onto an

A: First of all, to define a bijection from [a, b] to [c, d], the condition d-c=b-a must be satisfied.…

Q: 2 5= {(0, 0₂, 93) | DER₁, A₂ER₂, az ER₂ Z Show that sis subring of RxP₂Rz

A: We have to show that S is a subring of ring R1×R2×R3.

Q: 2. Let R and S be equivalence relations. Prove that RnS is an equivalence relation.

A: Let A be a non-empty set. A relation R on A is said to be equivalence relation if it is reflexive…

Q: (7.8) Lemma. Let K, L be simplicial complexes which intersect in a common subcomplex, then z(K U L)…

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Q: Theorem: if I is a Positive implicative Bck-algebra Then (Lx),,Lo) is a BCK-algebra

A: Consider the given theorem, '' if X is positive implicative Bck-algebra then Lx,⊙,L0 is a…

Q: 5. Let (1 2 3 A = 2 1 4 1 0 2 Find A-lif possible. If A-does not exist, explain why.

A: Given See the handwritten solution

Q: Let G=U(16),H={1,15},andK={1,9} . Are H and K isomorphic?Are G/H and G/K isomorphic?

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Q: (d) Is T an isomorphism?

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Q: Show that the power set of 2, 2º, is algebra on 2. an

A: A σ-algebra is a type of algebra so, every sigma algebra is an algebra.        (1)

Q: Prove that Z[√ -2] and Z[√ 2] are unique factorization domains.

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Q: Show that U(8) is isomorphic to U(12).

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Q: Show that Z4 is not isomorphic to U(12) by comparing the orders of elements.

A: z4=0,1,2,3 is group w.r to +order of elementsoa=n⇒an=e where n is least positive…

Q: Find the orders of all the elements of Ag.

A: A8 is the group of all even permutation of S8.

Q: 1. Prove Theorem 1 by setting up an isomorphism between (n + 1)-tuples and poly- nomials of degree…

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Q: Build an isomorphism between M2  (R) and P3  (R)

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Q: Let a ER which expressions are part of the Borel o algebra, B(R) they are correct? and give the…

A: The main objective is to determine the correct option.

Q: 7. Determine whether g(r) =5 6x even, odd or neither. Discuss its symmetry. -

A: f(x) = f(-x), then f(x) is even function,  f(x) not equal to f(-x), then f(x) is odd function.

Q: Prove that the intersection of any number of o-algebras is a o-algebra.

A: In mathematical analysis and in probability theory, a σ-algebra on a set X is a collection \Sigma of…

Q: Abstract Algebra: Prove that Aut(Z/18Z) is cyclic.

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Q: 3. Construct a bijection f: [0, 1] → (0, 1) UN.

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Q: 4. Assuming that gcd(a, b) = 1, prove the following: (a) gcd(a +b, a – b) = 1 or 2. %3D %3D

A: As per our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: U(8) is isomorphic to Z4. Select one: True False

A: U(8) and Z4 both have the same elements of order 4 . Now if there is an isomorphism between U(8) and…

Q: Prove, by comparing orders of elements, that Z8 ⨁ Z2 is not isomorphicto Z4 ⨁ Z4.

A: Given  Z8 ⨁ Z2 and Z4 ⨁ Z4.

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

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Q: Show that for A and B of n×n order with complex enteries the matric Ab need not be similar to BA ,…

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Q: Prove that vn e Z, gcd(7n + 1, 15n + 2) = 1 *gcd = Greatest common divisor

A: Below is the solution for the above question.

Q: Show that there is no associative algebra A such that the Lie algebra arising from A is sl2(C).

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Q: a) Let * be the operation on Z defined by a*b= a + b - 3ab for all a,b e Z. Justify the following…

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Q: (B): Let M be simple R -module , show if M is cyclic or not? (why)

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Q: 1 (a) Show using set identities that A UBU (AN Bn C) = AUBUC (b) Prove by contradiction that if A C…

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Q: is union of two ideal rings R, an idea of R? prove or give counter example

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Q: Exercise 3.2.3 Show that Zg under addition is isomorphic to Z; under multiplication.

A: We know that Z6 = {0, 1, 2, 3, 4, 5} and Z7* = {1, 2, 3, 4, 5, 6}. A group isomorphism is a function…