Question

Let G be a group and α : G → H be a homomorphism of groups...

Let G be a group and α : G → H be a homomorphism of groups with H abelian. Show that α factors via G/[G, G], i.e. there exists a homomorphism β : G/[G, G] −→ H, such that α = β◦q, where q : G −→ G/[G, G] is the quotient homomorphis

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
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.
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 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 N and H be groups, and here for a homomorphism f: H --> Aut(N) =...
Let N and H be groups, and here for a homomorphism f: H --> Aut(N) = group automorphism, let N x_f H be the corresponding semi-direct product. Let g be in Aut(N), and  k  be in Aut(H),  Let C_g: Aut(N) --> Aut(N) be given by conjugation by g.  Now let  z :=  C_g * f * k: H --> Aut(N), where * means composition. Show that there is an isomorphism from Nx_f H to Nx_z H, which takes the natural...
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 H be a normal subgroup of G. Assume the quotient group G/H is abelian. Prove...
Let H be a normal subgroup of G. Assume the quotient group G/H is abelian. Prove that, for any two elements x, y ∈ G, we have x^ (-1) y ^(-1)xy ∈ H
4. Let f : G→H be a group homomorphism. Suppose a∈G is an element of finite...
4. Let f : G→H be a group homomorphism. Suppose a∈G is an element of finite order n. (a) Prove that f(a) has finite order k, where k is a divisor of n. (b) If f is an isomorphism, prove that k=n.
Let G be an Abelian group and H a subgroup of G. Prove that G/H is...
Let G be an Abelian group and H a subgroup of G. Prove that G/H is Abelian.
Let G be a group and let p be a prime number such that pg =...
Let G be a group and let p be a prime number such that pg = 0 for every element g ∈ G. a.      If G is commutative under multiplication, show that the mapping f : G → G f(x) = xp is a homomorphism b.     If G is an Abelian group under addition, show that the mapping f : G → G f(x) = xpis a homomorphism.
Let G be an abelian group, let H = {x in G | (x^3) = eg},...
Let G be an abelian group, let H = {x in G | (x^3) = eg}, where eg is the identity of G. Prove that H is a subgroup of G.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT