Question

Let G and H be groups, and let G0 = {(g, 1) : g ∈ G}...

Let G and H be groups, and let

G0 = {(g, 1) : g ∈ G} .

(a) Show that G0 ≅ G.

(b) Show that G0 is a normal subgroup of G × H.

(c) Show that (G × H)/G0 ≅ H.

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 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 G and G′ be two isomorphic groups that have a unique normal subgroup of a...
Let G and G′ be two isomorphic groups that have a unique normal subgroup of a given order n, H and H′. Show that the quotient groups G/H and G′/H′ are isomorphic.
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 be the group Z3 + Z4 and let H = h(1, 2)i be the...
Let G be the group Z3 + Z4 and let H = h(1, 2)i be the cyclic subgroup generated by (1, 2). (a) Find the index [G : H] of H in G. (b) Is H a normal subgroup of G? Justify your answer.
Let G be a finite group and let H be a subgroup of order n. Suppose...
Let G be a finite group and let H be a subgroup of order n. Suppose that H is the only subgroup of order n. Show that H is normal in G. Hint: Consider the subgroup aHa-1 of G. Please explain in detail!
Let H be a subgroup of G, and N be the normalizer of H in G...
Let H be a subgroup of G, and N be the normalizer of H in G and C be the centralizer of H in G. Prove that C is normal in N and the group N/C is isomorphic to a subgroup of Aut(H).
prouve that let H normal subgroup of G and (|G:H|,|H|)=1 H hall subgroup then H characterstic...
prouve that let H normal subgroup of G and (|G:H|,|H|)=1 H hall subgroup then H characterstic G
Let H, K be two groups and G = H × K. Let H = {(h,...
Let H, K be two groups and G = H × K. Let H = {(h, e) | h ∈ H}, where e is the identity in K. Show that G/H is isomorphic to K.
Suppose H is a normal subgroup of G where both H and G/H are solvable groups....
Suppose H is a normal subgroup of G where both H and G/H are solvable groups. Prove that G is then a solvable group as well.
Let G be a group and suppose H = {g5 : g ∈ G} is a...
Let G be a group and suppose H = {g5 : g ∈ G} is a subgroup of G. (a) Prove that H is normal subgroup of G. (b) Prove that every element in G/H has order at most 5.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT