Abstract Algebra help needed ASAP or doomed

[cj.art1]

I need help with these problems ASAP. These are from a practice test.

1. Suppose |G| = pq, where p and q are primes. Prove that either G ? Zp × Zq or Z(G) = {e}.
2. Give an example of a group G and its subgroup H of index 2008 that is not normal.
3. Prove that Z(G1 × G2) ? Z(G1) × Z(G2).
4. Let H be a subgroup of Sn. Prove that either H ? An or exactly half of elements of H are in Sn.
5. Suppose H is a normal subgroup of G of index 7 and K is a subgroup of G of order 5. Prove that K is contained in H.
6. Prove that (Z × Z)/h(1, 2)i is isomorphic to Z.
7. Describe all normal subgroups in S3 × Z2.
8. Let H ? Z × Z be a subset of all pairs (a, b) such that a + b is even. Prove that H is a subgroup and compute its index in Z × Z.
9. Describe all homomorphisms from Z4 to D4 and from D4 to Z4.
10. Prove that Inn(G × Z2) ? Inn(G) for any group G.
11. Prove that if G1 × G2 is cyclic then G1 and G2 are both cyclic, both finite, and their orders are coprime.
12. Prove that Z2 is the only finite group that has no nontrivial automorphisms.
13. Let H,K be subgroups of an Abelian group G and suppose there exists a homomorphism  p: G ? H such that p(x) = x for any x ? H and Ker p = K. Prove that there exists a homomorpism r: G ? K such that r(x) = x for any x ? K and Ker r = H.

 

[mmm4444bot]

Hello CJ:

I think you're doomed, because I speculate that not many people will take time out to explain much of this to you from scratch.

But, look on the bright side. Three of these exercises do not involve proofs.

I wish you good fortune,

~ Mark :|

PS: Read the post titled "Read Before Posting" to learn how to accommodate potential helpers at this site. If you can let them know what you know about these exercises, and what you've tried, or why you're unsure or stuck, then they will have a much better idea of what kind of help you need. If they don't know why you're stuck, then they might not respond at all. Alternately, you can wait for Fast Eddie and take your chances.

MY EDIT: retracted exaggeration

 

[cj.art1]

Eeep, I'm not looking for all the answers from the same person, just if someone can contribute and do one of these offhand, it would help!

Edit: I understand what a group, subgroup, identity, 'normal', abelian, symmetires of S, and kind of what an isomorphism and index are. I'm still confused about order. I don't know what a homomorphism is, and i've tried but failed to understand what cyclic and coprime are. I have no idea what Inn means or the vector <(1,2)>. I still don't get Galois's Theorem or Sylow's theorem.

 

[mmm4444bot]

... I'm not looking for all the answers ...

Thank you for clarifying; intentions and motivations are not always clear with these types of posted exercise lists. By the way, when you look at your post, do you see all of the symbols?

I'm seeing several empty squares where I think some symbols should be.

~ Mark :?

 

[stapel]

if someone can contribute and do one of these offhand, it would help!

I'm sorry, but I don't understand...?

You already have all the examples and explanations in your textbook, your class notes, and whatever other resources you have accessed (such as the various online lessons you've studied). How would one more worked example make the difference? :shock:

Eliz.