List all the homomorphisms from D5 to Z2 + Z2.

Q: Construct a homomorphism of rings p:Z[i] → Z,

A: Consider the rings ℤi and ℤ2. Define a map φ:ℤi→ℤ by φa+ib=0 ∀ a,b∈ℤ. Let a+ib, c+id∈ℤi.…

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: a с T: R2 [x] → M2x2 (R) is given by T(a + bæ + ca²) = (C Which b b+c-a statement about T is…

A: Linear Transformation

Q: 0:C' →C',o(x) = x*ANYWAY. O SOW THAT THERE IS A HOmomoRPHISm AND KER(Ơ) DETERMINE

A: Group Homomorphism: A Homomorphism ϕ from a group G, ∘ to a group G¯, ⊛ is a mapping from Ginto G…

Q: Compute Ker(ø) for the homomorphism ø: Z→Z, such that (1)=9 6Z 8Z O None of them 5Z 7Z

A: Given∅(1) = 9 then ∅(2) = 2∅(1)= 2(9)= 18= 3 mod 15= 3 Hence ∅(2) = 3 Also ∅(3) = 3∅(1)= 3(9)= 27=…

Q: Let P3 have the inner product given by evaluation at -7, -1, 1, and 7.

A: Given, ℙ3 have the inner product given by evaluation at -7, -1, 1 and 7. And P0(t)=3, p1(t)=2t,…

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

A:

Q: Find Aut(Z6).  Define each automorphism and justify your conclusion.

A: Concept:

Q: 2. If f: A B is an equivalence in a category e and g : B A is the morphism such that gof= 14, fo g =…

A:

Q: Let ficxT)→(メッテ)and g: メっテ)→(Kっズ) e are homomorphism then gof 1s homo.

A:

Q: Without looking at inner automorphisms of Dn, determine the numberof such automorphisms

A:

Q: Determine all ring homomorphism from Z6 to Z6

A: Z6 is a cyclic, f is completely determined by f(1).

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

A:

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: Is U (8) ∼= Z2 ×Z2 ? why? If yes, gives an explicit isomorphism.

A:

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 ring homomorphism from Zn to Zn

A: (if n is not prime) No. of ring homomorphism from Zn to Zn = No. of Idempotent elements in Zn.

Q: Determine all homomorphisms from Z4 to Z2 ⨁ Z2.

A:

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

A:

Q: Determine all ring homomorphisms from Z to Z.

A: In algebra, A homomorphism is defined as the similarity between the shape, structure, form, etc…

Q: 47. Show that 0: C → M2x2(R) which is given a by 0(a + ib) = ( ) for all a, b eR is an isomorphism.

A: To prove isomorphism, θ should be one-one and onto One-one θa1+ib1=θa2+ib2impliesa1+ib1=a2+ib2…

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: Q) Make up a Kelly table for set G= [1,-1, i,-i} with respect to multiptication and then explain…

A:

Q: Show that the mapping :S3 S3 given by p(x) = xis not a homomorphism.

A: Solution is below.

Q: 15. Let x = (x1, x2). Using the feature mapping show that *(2,3)·¤((4,4) –-(2,3)·(4,4)*

A: Given that for x=x1,x2, the feature mapping is…

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

A:

Q: (d) Is T an isomorphism?

A:

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

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

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

A:

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).

A:

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: Which of the following are homomorphism? Monomorphism? Epiomorphism? Isomorphism? 1/ f: (Z, +)…

A:

Q: Describe all extensions of the identity map of ℚ to an isomorphism mapping ℚ(3rd root of 2) onto a…

A:

Q: State and prove the first isomorphism theorem.

A: As per the guidelines we are supposed to solve the first questions only.kindly post another…

Q: What is the kernel of the homomorphism : Z4 → Z10 defined by ø(x) = 5x? {0, 2} {0, 2, 4, 6, 8} {1,…

A:

Q: Every continuous map T2 –→ S2 is homotopic to the constant map

A: To prove: Every continuous map from the Torus T2 to the sphere S2 is homotopic to the constant map.

Q: Find all ring homomorphisms from R to Z5.

A:

Q: Which sequence of transformations maps ABCD onto A'B'C'D' and then maps A'B'C'D' onto A"B"C"D"?

A: Answer is option (1).

Q: o is a) Homomorphism. b) Isomorphism c) Automorphism d) None

A:

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: Let f: MM' be an R-module homomorphism, then

A:

Q: Which of the following is not an automorphism? * Ø: (R*,.) (R*,.) with 0(x) = Vx 0: (R*,.) (R +)…

A:

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

A:

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

A:

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

A:

Q: prove or disprove : If F:(R,+) (R\{0},.), defined by f(x)= x +2 is an isomorphism.

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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…