Group Theory Final Study Guide 1

So, I'll be posting study guides for Group Theory and PHL in chunks over the next few days. I'll also be posting some problems from the Discrete Math review sheet. What can I say? It's crunch time, folks...
I'll be expected to know the definitions of the following terms for my Group Theory final:

1. Group
2. Order of an element
3. Factor group
4. Group homomorphism
5. Kernel of a homomorphism

Ok, here goes. Keep in mind, this is from memory as much as possible.

1. Group: A group G is a set with a binary operation that has the following characteristics: 1) It is transitive; that is, for all a,b,c in G, (ab)c=a(bc). 2) There is an identity element e, such that, for all a in G, ae=ea=a. 3) There are inverses; for all all a in G, there is an element b in G such that ab=ba=e.
   
2. Order of an element: In a group G with element g, the order of g is the smallest positive integer n such that g^n=e.
   
3. Factor group: Define a group G with a normal subgroup H. The factor group G/H is defined as G/H={aH| a in G}, under the operation (aH)(bH)=abH. What does this mean? It means that G/H is the set of all (distinct) cosets of H in G, under the specified operation. It should be noted that the orders |aH| and |a| may not be equal. In fact, even though H contains e, the identity of G/H is NOT e, but, rather H (specifically, it is eH). Further, note that (aH)^2=(a^2)H, NOT (a^2)(H^2). Just remember that G/H is a set of (distinct!) cosets of H in G under the operation aH*bH=abH. And breath deep.
   
4. Homomorphism: A homomorphism P on a group G is a mapping from G into G' that preserves operation; that is, for all a,b in G, P(ab)=P(a)P(b). A few things to note: P maps INTO not ONTO; rather, not necessarily onto.
   
5. Kernel of a homomorphism: Define a group G and a homomorphism P that maps G to G' with the identity element e. Then, the kernel of P is KerP={x in G| P(x)=e}; that is, the kernel is the set of all elements in G that are mapped to the identity of G' by P.
   

Here are some things to remember about homomorphisms:

1. P carries the identity of G to G'.
   
2. P(g^n)=(P(g))^n for all n in Z. This means that...
   
3. If |g| is finite, then |P(g)| divides |g|.
   
4. KerP is a subgroup of G.
   
5. P(a)=P(b) iff aKerP=bKerP. I'll come back to this one, I think...
   
6. If P(g)=g', then Pinv(g')={x in G| P(x)=g'}=gKerP. This means that Pinv(g') is a SET, not necessarily just one element. Also, note that P(KerP)=e and P(gKerP)=g'. Interesting...
   
7. If H is a subgroup of G, then P(H) is a subgroup of G'.
   
8. If H is cylic, then P(H) is cylic.
   
9. If H is Abelian, then P(H) is Abelian.
   
10. If H is normal, then P(H) is normal.
    
11. The order of P(H) divides the order of H.
    
12. KerP is always normal to G.
    
13. If P is onto and KerP={e}, then P is an isomorphism from G onto G'.
    

Here are some things to remember about factor groups:

1. If G/Z(G) is cyclic, then G is Abelian.
   
2. G/Z(G) is isomorphic to Inn(G).
   
3. Cauchy's Theorem for Abelian Groups: Let G be a finite Abelian group and let p be a prime that divides the order of G. Then G has an element of order p.
   

That's all for the moment. The super important things to recall will be the properties of homomorphisms and Cauchy's Theorem. That's a lot of properties to recall about homomorphisms.
I'll post some proofs on Sunday, and some properties and examples and such on Monday or Tuesday. Also, expect some problems from Discrete later tonight.
Ciao.