Question

Let φ : G → G′ be an onto homomorphism and let N be a normal...

Let φ : G → G′ be an onto homomorphism and let N be a normal subgroup of G. Prove that φ(N) is a normal subgroup of G′.

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
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.
Let C be a normal subgroup of the group A and let D be a normal...
Let C be a normal subgroup of the group A and let D be a normal subgroup of the group B. (a) Prove that C × D is a normal subgroup of A × B (b) Prove that the map φ : A × B → (A/C) × (B/D) given by φ((m, n)) = (mC, nD) is a group homomorphism. (c) Use the fundamental homomorphism theorem to prove that (A × B)/(C × D) ∼= (A/C) × (B/D)
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
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
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}.
Suppose G, H be groups and φ : G → H be a group homomorphism. Then...
Suppose G, H be groups and φ : G → H be a group homomorphism. Then the for any subgroup K of G, the image φ (K) = {y ∈ H | y = f(x) for some x ∈ G} is a group a group in H.
Let φ:G ——H be a group homomorphism and K=ker(φ). Assume that xK=yK. Prove that φ(x)=φ(y)
Let φ:G ——H be a group homomorphism and K=ker(φ). Assume that xK=yK. Prove that φ(x)=φ(y)
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.
Letφ:G→G′is a group homomorphism. Prove that φ is one-to-one if and only if Ker(φ) ={e}.
Letφ:G→G′is a group homomorphism. Prove that φ is one-to-one if and only if Ker(φ) ={e}.
Let G be a group and a be an element of G. Let φ:Z→G be a...
Let G be a group and a be an element of G. Let φ:Z→G be a map defined by φ(n) =a^{n} for all n∈Z. Find the image φ(Z) and prove that φ(Z) a subgroup of G
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT