Suppose that Φ is a homomorphism from U(40) to U(40) and thatKer Φ = {1, 9, 17, 33}. If Φ(11) = 11, find all elements of U(40)that map to 11.

Q: Prove that the mapping from G O H to G given by (g, h) → g is a homomorphism. What is the kernel?…

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Q: 1- Which of the following is a homomorphism? a)O f:Z→Z, f(x) = x+2 %3D b) O f: Z12 → Z30, ƒ(x) = 20x…

A: Consider a function    .Then f is a homomorphism if and only if  . .

Q: the only homomorphism f  : S6 to  Z77 is the trivial one,  Select one: True False

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Q: (3) Let (A, +..) be a subgroup of (M₂ (Z), +,.), Then A is ideal of M₂ (Z), where A = {(a b) la, b,…

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Q: Which of the following is not an isomorphism? * f: (R", )→ (R, +) where f(x) = Inx %3D O this option…

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Q: If Ø: R S is a ring homomorphism. The Ø preserves:

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Q: Suppose that Φ is a homomorphism from D12 onto D3. What isΦ(R180)?

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Q: If any two members P and Q of a group satisfy the relation P = Q'PQ, then P and Q commute. Select…

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Q: ) Every continuous map RP2 → S² is homotopic to the constant map.

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Q: Let S, be the set of all one-one and onto mappings on {1,2,3} and Z, = (0, 1,2, 3, 4}. Then, show…

A: Let G be a group and ∗ be an operation defined in G. Then (G, ∗) is group if it satisfies following…

Q: Suppose that o is a homomorphism from Z3, to Z30 and Ker ø = {0, 10, 20}. If (23) = 9, determine all…

A: Given: ɸ is a homomorphism from Z30 to Z30. Ker ɸ = {0, 10, 20}.

Q: ) If we have an epimorphism from G to G’, then we know G must be the same or of bigger order than…

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Q: Let G be a group and let r, y e G such that ya = r-ly. Use the Principle of Mathematical Induction…

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Q: Find all homomorphisms from Zg →Z10 , and state the kernel of each of these homorphisms.

A: We have to find all the homomorphisms from ℤ8 to ℤ10 , and we have to state the kernel of each of…

Q: When the preimage and image are congruent, the transformation is called an isometry. O True O False

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Q: 1) Prove that G = {a + bv2 \a,b € Q and a, b # 0} is a subgroup of (R – (0},.)

A: Subgroup

Q: 2. If a is an involution, then Baß is an involution for any transformation B.

A: It is given that α is an involution. So, we can say that α2=I.

Q: Which of the following is isomorphism? * f: (R*, .)(R, +) where f(x) = Inx f: (Z, +) → (R*, .) where…

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Q: For the group homomorphism : Zs → Z, defined by ø([r]) = [æ]² for all [r] € Z, find the kernel and…

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Q: 8) Determine whether the following graphs are isomorphic. If they are, state the isomorphism f and…

A: Here, we will be showing the given two graphs are not isomorphic. We are given the graphs,

Q: Which of the following is not an automorphism? * Ø: (Z, +) → (Z, +) with Ø(x) = 3x %3D This Option…

A: Use the definition of Automorphism to solve this problem.

Q: Show that the set A = {fm,b : R → R | m # 0 and fm,b(æ) group under composition of functions. (The…

A: To Show- Show that the set A = fm,b : R → R | m ≠ 0 and fm,bx = mx + b, m, b ∈ R forms a group under…

Q: 23. Let 0:→ be defined by a b = a+d (a,b,c,de R ) a) Prove that 0 is a homomorphism b) Determine…

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Q: * If f.g:G → G'beHomomorphism of G to G* then show that gof : G →G' will be also Homomorphism of G…

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Q: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a^2= e a =…

A: Since f is a group homomorphism , where f(x)=a∗x∗a−1,    x∈G. So a^-1=a implies self inverse implies…

Q: An onto grouphomomorphisimf: G H is an isomorpisim if and only if Select one: a. f(1,)=H b.…

A: We have to solve given problem:

Q: Can you write a group homomorphism as φ (gh) as φ(hg)? Are they the same thing?

A: The given homomorphism ϕgh, ϕhg The objective is to find whether the ϕgh,ϕhg are same.

Q: express Z10 as a product of Z5 x Z2 , verify that both groups are isomorphic.

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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: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a = e O a^4…

A: Given that f:G→G be a function such that fx=axa

Q: E. Let o : R+ → R* under multiplication be given by o(x) = |x|. 1. Show that is a homomorphism. 2.…

A: ϕ: R+→R+ϕ(x)=x

Q: 2.3 Let 9: (M2(R), +)→ be defined by e c d a b = a +d. where a,b,c,d €R (a) Prove that e is a…

A: A mapping f:(G,+)→(H,+) is said to be group homomorphism if it satisfies: f(a+b)=f(a)+f(b)…

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

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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: Prove that H= { |ne Z} is a subgroup of GL2(R) under multiplication.

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Q: What is a mapping of phi defined by the automorphism of Z6?

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Q: Which of the following is isomorphism? f:(Z, +) → (Z, +) where f(x) = 2x. this option f: (R, +) →…

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Q: Prove that T is a lincar transformation.

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Q: Which of the following is isomorphism? f: (Z, +) > (Z, +) where f(x) = 2x. O None of these this…

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Q: the only homomorphism f : S, to Z,, is the trivial one, Select one: O True O False

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Q: Find the group homomorphism between (Z, +) and (R- (0},.)

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Q: Define p: Z16 Z16 by p(x) = 4x is a homomorphism then Kerø = a) {0,12} b) {0,4,8,12} c) {4,8,12} d)…

A: Given  ϕ:ℤ16→ℤ16  by ϕ(x)=4x is a homomorphism. To find: Ker ϕ

Q: Suppose that f:G - G such that f(x) = axa". Then fis a group homomorphism if and only if O a^3 = e a…

A: f:G→G such that  fx=axa2 We know that f is a homomorphism if fxy=fxfy for all x, y∈G

Q: 24. Let G = s C. Define 0: - by o( n) =i" a) Verify that o is a homomorphism b) Find Ker( o )

A: Apply Homorphism definition

Q: If f: Sn--- Sn is a homomorphism, prove tha is a subset of An.

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Q: Suppose that is an isomorphism from Z, O Z, to Z5 and $(2, 3) = 2. Find the element in Z, Z, that…

A: given, ϕ is an isomorphism from ℤ3⊕ℤ5 to ℤ15  and ϕ2,3=2

Q: Show that linear mapping  T : P2 (R) → R3defined by  T (ax2 + bx + c) = (a + b, a + c, b+ c), is an…

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Q: Consider p: IR → R under addition, defined by p(x) = x2. a) o is a homomorphism b) o is not a…

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Q: P: Sn → Z/2Z (0, if o E An (1, if o ¢ An P(0) Check to see if o is a homomorphism? Is p an…

A: It is given that φ:Sn→ℤ2ℤ φσ=0, if σ∈An1, if σ∉An Homomorphism:  φσ1·σ2=φσ1·φσ2 We have φσ1=0, if…