Question

A) Prove that a group G is abelian iff (ab)^2=a^2b^2 fir any two ekemwnts a abd...

A) Prove that a group G is abelian iff (ab)^2=a^2b^2 fir any two ekemwnts a abd b in G.
B) Provide an example of a finite abelian group.
C) Provide an example of an infinite non-abelian group.
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 be a group (not necessarily an Abelian group) of order 425. Prove that G...
Let G be a group (not necessarily an Abelian group) of order 425. Prove that G must have an element of order 5. Note, Sylow Theorem is above us so we can't use it. We're up to Finite Orders. Thank you.
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 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
For an abelian group G, let tG = {x E G: x has finite order} denote...
For an abelian group G, let tG = {x E G: x has finite order} denote its torsion subgroup. Show that t defines a functor Ab -> Ab if one defines t(f) = f|tG (f restricted on tG) for every homomorphism f. If f is injective, then t(f) is injective. Give an example of a surjective homomorphism f for which t(f) is not surjective.
Let a,b be any two elements of a group. Prove that' (ab)-1 = b-1a-1
Let a,b be any two elements of a group. Prove that' (ab)-1 = b-1a-1
Let G be a simple graph with n(G) > 2. Prove that G is 2-connected iff...
Let G be a simple graph with n(G) > 2. Prove that G is 2-connected iff for every set of 3 distinct vertices, a, b and c, there is an a,c-path that contains b.
Let G be a group. Prove that following statements are equivalent. a.) G is commutative b.)...
Let G be a group. Prove that following statements are equivalent. a.) G is commutative b.) ∀ a,b ∈ G, (ab)2 = a2b2 c.) ∀ n ∈ N, ∀ a,b ∈ G, (ab)n = anbn
(abstract alg) Let G be a cyclic group with more than two elements: a) Prove that...
(abstract alg) Let G be a cyclic group with more than two elements: a) Prove that G has at least two different generators. b) If G is finite, prove that G has an even number of generators
Let G be a non-abelian group of order p^3 with p prime. (a) Show that |Z(G)|...
Let G be a non-abelian group of order p^3 with p prime. (a) Show that |Z(G)| = p. (b) Suppose a /∈ Z(G). Show that |NG(a)| = p^2 . (c) Show that G has exactly p 2 +p−1 conjugacy classes (don’t forget to count the classes of the elements of Z(G)).
3. a) For any group G and any a∈G, prove that given any k∈Z+, C(a) ⊆...
3. a) For any group G and any a∈G, prove that given any k∈Z+, C(a) ⊆ C(ak). (HINT: You are being asked to show that C(a) is a subset of C(ak). You can prove this by proving that if x ∈ C(a), then x must also be an element of C(ak) for any positive integer k.) b) Is it necessarily true that C(a) = C(ak) for any k ∈ Z+? Either prove or disprove this claim.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT