Question

Let ϕ : Z × Z → Z be a homomorphism such that ϕ ( 1...

Let ϕ : Z × Z → Z be a homomorphism such that ϕ ( 1 , 1 ) = 2 and ϕ ( 3 , 5 ) = 6 . Find Ker ϕ and ϕ （10，5）

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Prove this statement: Let ϕ : A1 → A2 be a homomorphism and let N =...
Prove this statement: Let ϕ : A1 → A2 be a homomorphism and let N = ker ϕ. Then A1/N is isomorphic to ϕ(A1). Further ψ : A1/N → ϕ(A1) defined by ψ(aN) = ϕ(a) is an isomorphism. You must use the following elements to prove: - well-definedness - one-to-one - onto - homomorphism
Suppose G and H are groups and ϕ:G -> H is a homomorphism. Let N be...
Suppose G and H are groups and ϕ:G -> H is a homomorphism. Let N be a normal subgroup of G contained in ker(ϕ). Define a mapping ψ: G/N -> H by ψ (aN)= ϕ (a) for all a in G. Prove that ψ is a well-defined homomorphism from G/N to H. Is ψ always an isomorphism? Prove it or give a counterexample
Let G and H be groups and f:G--->H be a surjective homomorphism. Let J be a...
Let G and H be groups and f:G--->H be a surjective homomorphism. Let J be a subgroup of H and define f^-1(J) ={x is an element of G| f(x) is an element of J} a. Show ker(f)⊂f^-1(J) and ker(f) is a normal subgroup of f^-1(J) b. Let p: f^-1(J) --> J be defined by p(x) = f(x). Show p is a surjective homomorphism c. Show the set kef(f) and ker(p) are equal d. Show J is isomorphic to f^-1(J)/ker(f)
Let φ : A → B be a group homomorphism. Prove that ker φ is a...
Let φ : A → B be a group homomorphism. Prove that ker φ is a normal subgroup of A.
Suppose that ϕ:R→S is a ring homomorphism and that the image of ϕ is not {0}....
Suppose that ϕ:R→S is a ring homomorphism and that the image of ϕ is not {0}. If R has a unity and S is an integral domain, show that ϕ carries the unity of R to the unity of S.
Let G be a cyclic group, and H be any group. (i) Prove that any homomorphism...
Let G be a cyclic group, and H be any group. (i) Prove that any homomorphism ϕ : G → H is uniquely determined by where it maps a generator of G. In other words, if G = <x> and h ∈ H, then there is at most one homomorphism ϕ : G → H such that ϕ(x) = h. (ii) Why is there ‘at most one’? Give an example where no such homomorphism can exist.
Please explain it in detail. Let φ∶G → H be a homomorphism with H abelian. Show...
Please explain it in detail. Let φ∶G → H be a homomorphism with H abelian. Show that G/ ker φ must be abelian.
Let G be an Abelian group. Let k ∈ Z be nonzero. Define φ : G...
Let G be an Abelian group. Let k ∈ Z be nonzero. Define φ : G → G by φ(x) = x^ k . (a) Prove that φ is a group homomorphism. (b) Assume that G is finite and |G| is relatively prime to k. Prove that Ker φ = {e}.
Let α = 4√ 3 (∈ R), and consider the homomorphism ψα : Q[x] → R...
Let α = 4√ 3 (∈ R), and consider the homomorphism ψα : Q[x] → R f(x) → f(α). (a) Prove that irr(α, Q) = x^4 −3 (b) Prove that Ker(ψα) = <x^4 −3> (c) By applying the Fundamental Homomorphism Theorem to ψα, prove that L ={a0+a1α+a2α2+a3α3 | a0, a1, a2, a3 ∈ Q }is the smallest subfield of R containing α.
Prove the following theorem: Let φ: G→G′ be a group homomorphism, and let H=ker(φ). Let a∈G.Then...
Prove the following theorem: Let φ: G→G′ be a group homomorphism, and let H=ker(φ). Let a∈G.Then the set (φ)^{-1}[{φ(a)}] ={x∈G|φ(x)} =φ(a) is the left coset aH of H, and is also the right coset Ha of H. Consequently, the two partitions of G into left cosets and into right cosets of H are the same
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT