Study Guide 1 for Ring and Field Theory, Midterm 2

So it's midterm time again. Here are some definitions I'll need to iterate on the exam. Later, I might put up some proofs. Honestly, I've found the material this term much easier than last term, so I haven't felt the need to blog as much. Sorry.

• Principal Ideal Domain (PID): An integral domain D is a PID if every ideal is of the form 'a'={ar| r in D}.
• Ring of Polynomials: Let R be a commutative ring. The set of formal symbols R[x]={a(n)x^n+a(n-1)x^(n-1)+...+a(1)x+a(0)| a(i) in R, n is a nonnegative integer} is called the ring of polynomials over R in the indeterminate x. Two elements, a(n)x^n+a(n-1)x^(n-1)+...+a(1)x+a(0) and b(m)x^m+b(m-1)x^(m-1)+...+b(1)x+b(0), are said to be equal iff a(i)=b(i) for all nonnegative integers i. Further, define a(i)=0 when i>n and b(i)=0 when i>m.
• Content of a Polynomial, Primitive Polynomial: In a polynomial ring R[x], the content of the element a(n)x^n+a(n-1)x^(n-1)+...+a(1)x+a(0) is the greatest common divisor of the coefficients, a(i). The element a(n)x^n+a(n-1)x^(n-1)+...+a(1)x+a(0) is said to be primitive if the content is 1 (that is, if at least one coefficient is relatively prime to all the others).
  
• Irreducible Polynomial: Let D be an integral domain. A polynomial f(x) from D[x] that is neither the zero polynomial nor a unit is said to be irreducible if, when f(x)=g(x)h(x), where g(x) and h(x) are in D[x], g(x) or h(x) is a unit in D[x].
  
• Prime Element: Let D be an integral domain, and let a, b, c be in D. The nonzero, non-unit element a is prime if, if a|bc then a|b or a|c.
  
• Vector Space: A set V is said to be a vector space over a field F is V is an Abelian group under addition, and, if for each a in F and v in V, there is an element av in V such that the following conditions hold for all a,b in F and all u,v in V: 1) a(v+u)=av+au; 2) (a+b)v=av+bv; 3) a(bv)=(ab)v; 4) 1v=v.
  
• Linear Independence: A set of vectors S is said to be linearly independent over a field F if there are vectors v1, v2,..., vn from S and elements a1, a2,...,an from F, not all zero, such that a1v1+a2v2+...+anvn=0.
  

Ok, those are the definitions I need to know. Pretty straightforward. The vector space definition is a little long winded, as is the polynomial ring definition, but I'll have to make do... Until proof time, folks.

Proofs and Theorems for Exam

Here are some proofs that I may need iterate on the exam. Following that are some theorems that I'll probably need to know.

• Claim: A finite integral domain is a field.
  Pf: Let D be a finite integral domain with unity 1. Let a be any nonzero element in D. I will show that a is a unit. If a=1, it is its own inverse, so I am done. So, assume that a =! 1. Then, let D contain a, a^2, a^3, a^4.... D is finite by assumption, so there must be two positive integers i and j such that i>j and a^i=a^j. Then, a^(i-j)*a^(j)=1*a^j. By cancellation, a^(i-j)=1. Since a=!1, i-j>1. Then, a*a^(i-j-1)=1. So, a^(i-j-1) is the inverse of a. qed.
  
• Claim: The characteristic of an integral domain is 0 or prime.
  Pf: It suffices to show that if the additive order of 1 is finite, it must be prime, because, in a ring with unity, the additive order of 1 is characteristic of the ring. Suppose that 1 has order n and that n=st, where 1<=s, t<=n. Then, 0=n*1=(st)*1=(s*1)*(t*1). There are no zero-divisors in an integral domain, so either s*1=0 or t*1=0. Since n is the least positive integer with the property that n*1=0, we must have s=n or t=n. Thus, n is prime. qed.
  
• Let R be a commutative ring with unity and let A be an ideal of R.
  Claim: R/A is an integral domain iff A is prime.
  Pf: Suppose that R/A is an integral domain and ab in A. Then, (a+A)(b+A)=ab+A=A. So, either a+A=A or b+A=A; that is, either a in A or b in A. Then, A is prime. To prove the converse, observe that R/A is a commutative ring with unity for any proper ideal A. Suppose that A is prime and that (a+A)(b+A)=0+A=A. Then, ab in A. Then, since A is prime, a in A or b in A. Then, a+A=A or b+A=A. That is, there can be no zero-divisors. qed.
  

There are two more that I am encouraged to know, but I don't think I'll need to know proofs. He generally only asks for proof of a theorem found in the book, but gives us three or four different options and lists five different theorems on the study guide. So, I'll be fine if I know these three.
Here are some theorems that I'll probably need to know:

• Cancellation: Let a, b, c be in an integral domain. If a=!0 and ab=ac, then b=c. NOTE: THIS IS NOT DIVISION.
  
• A finite integral domain is a field.
  
• For every prime p, Z(p) is a field.
  
• Let R be a ring with unity 1. If 1 has infinite additive order, then the characteristic of R is 0. Else, if 1 has order n, then the characteristic of R is n.
  
• The characteristic of an integral domain is 0 or prime.
  
• R/A is an integral domain iff A is prime.
  
• R/A is a field iff A is maximal.
  
• The kernel of any homomorphism is an ideal the ring.
  
• Every ideal A of a ring R is the kernel of a ring homomorphism r-->r+A from R to R/A.
  
• A ring with unity contains Z(n) or Z.
  
• A field contains Z(p) (if char=n) or Q (if char=0).
  

Ok, that's about all I need to know. Not bad, really. Its just the first midterm. Not bad at all.

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.

Group Theory Midterm Study Guide 3

Here's some examples of different types of groups, properties, and other bits of information that I might need to know for the exam.

Abelian Groups: A group G is Abelian iff ab=ba for all a,b in G.
Examples:

1. Z(n) under regular addition
   
2. U(n) under regular multiplication
3. Aut(Z(n)) under function composition

Properties:

1. Iff G is Abelian, then P(G) is Abelian.
   
2. If G is Abelian, then the subgroup H is necessarily Abelian.
   

Cyclic Groups: A group G is cyclic iff there exists an element a such that 'a'=G.
Examples:

1. In Z(n) with element k, if gcd(n,k)=1 (n,k relatively prime), then k is a generator of Z(n).
   
2. If |G| is prime, then G is cyclic.
   
3. Every subgroup of a cyclic group is cyclic.
   
4. For each positive divisor k of n, the set 'k/n' is the unique subgroup of Z(n) with order k; these are the ONLY subgroups of Z(n).
   
5. S(n) is cyclic.
   

Properties:

1. G is cyclic iff P(G) is cyclic.
2. Moreover, G='a' iff P(G)='P(a)'; that is to say, a generator maps to another generator.
   

Normal Groups: A subgroup H is normal to a group G iff xHx^-1=H, for all x in G.
Examples:

1. Any subgroup H of an Abelian group G is normal to G.
   
2. The centralizer Z(G) is always normal to G.
   
3. A(n) is always normal to S(n).
   
4. R(n) is always normal to D(n).
   
5. SL(2,R) is always normal to GL(2,R).
   

Isomorphisms: An isomorphism P is a mapping from a group G to a group P(G), where P is one-to-one, onto, and operation preserving.
Properties:

1. The identity always maps to the identity.
   
2. G is cyclic iff P(G) is cyclic.
   
3. G is Abelian IFF P(G) is Abelian.
   
4. G and P(G) have the same number of elements of each order, if G is finite.
   
5. G and P(G) have the same order.
   
6. |a|=|P(a)|; that is, P preserves the order of an element.
   
7. For a,b in G, a,b commute iff P(a),P(b) commute.
8. For all a in G, P(a^n)=(P(a))^n.
9. If K is a subgroup of G, then P(K) is a subgroup of P(G).
   
10. Aut(Z(n)) is isomorphic to U(n).
    

Cosets: Let H be a subset of a group G with element a; then, aH={ah| h is in H}. If H is a subgroup of G, then aH is a left coset of G. Similar definitions exists for Ha and aH(a^-1).
Properties:

1. a is in aH.
2. aH=H iff a is in H.
   
3. aH=bH iff a is in bH.
   
4. aH=bH or aH&bH={0}.
   
5. aH=bH iff (a^-1)b is in H.
   
6. |aH|=|bH|.
   
7. aH=Ha iff H=aH(a^-1).
   
8. aH is a subgroup of G iff a is in H.
   
9. The number of distinct cosets (left, right, OR whatever) of H in G is |G|/|H| (Lagrange's Theorem).
   

Lagrange's Theorem and Corollaries:
Lagrange's Theorem: Let G be a group with a subgroup H. Then, |H| divides |G|; also the number of distinct cosets of H in G is |G|/|H|.
Corollary 2: |a| divides |G|.
Corollary 3: All groups of prime order are cyclic.
Corollary 4: a^|G|=e.

External Direct Products: Let G1,..., Gn be a finite collection of groups. The external direct product of G1,..., Gn, written as G1#...#Gn, is the set of all n-tuples for which the ith component is an element of Gi and the operation is componentwise.
What That Means: Think of the Cartesian product. For example, Z(2)#Z(3)={(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)}, where (1,1)(1,2)=(0,0), because 1+1mod2=0 and 1+2mod3=0.
Properties:

1. The order of an external direct product is the product of the orders of the individual groups.
   
2. The order of an element of an external direct product is the least common multiple of the orders of the components of the element.
   
3. G#H is cyclic iff |G| and |H| are relatively prime.
   
4. Let m=n1*...*nk. Then, Z(m) is isomorphic to Z(n1)#...#Z(nk) iff ni and nj are relatively prime when i does not equal j. For example, Z(2)#Z(5)#Z(7)==Z(2)#Z(35)==Z(10)#Z(7)==Z(14)#Z(5)

This last property is still a little mysterious to me, but I'm coming around.
I need to go study for Discrete Math, now, but I might post some discrete stuff on here later.
Ciao, folks.
p.s.: I again apologize for the blatant lack of formality in the formatting, Deal with it.

Group Theory Midterm Study Guide 2

Proofs today, folks. I'll be proving the following theorems: Commutativity of Disjoint Cycles, Lagrange's Theorem (and the requisite 4th Property of the Lemma of Cosets), and the Corollaries of Lagrange's Theorem. Oh, just so there's no confusion, I'm not copying this out of my text; this is coming from memory (as much as possible) as an exercise for cementing these proofs for the exam (where I will be asked to give at least one of them).

Commutativity of Disjoint Cycles
Claim: If cycles A and B are disjoint, then AB=BA.
Pf: Define a set S such that S={a1,..., an, b1,..., bn, c1,..., cn}, where a's are elements mapped by A, b's are elements mapped by B, and c's are elements mapped by neither A nor B. Then, I need to show that (AB)(x)=(BA)(x) for all x in S. Let x=ai. Then, (AB)(ai)=A(B(ai))=A(ai)=a(i+1) and (BA)(ai)=B(A(ai))=B(a(i+1))=a(i+1). Now, let x=bi. Then, (AB)(bi)=A(B(bi))=A(b(i+1))=b(i+1) (BA)(bi)=B(A(bi))=B(bi)=b(i+1). Now, let x=ci. (AB)(ci)=A(B(ci))=A(ci)=ci and (BA)(ci)=B(A(ci))=B(ci)=ci. Then, (AB)(x)=(BA)(x) for all x in S. QED.

Property 4 of the Lemma of Cosets
Claim: aH=bH or aH&bH={0}.
Pf: Assume that aH and bH are left cosets of H in G. Let c be an element of aH and an element of bH. Property 3 of the Lemma of Cosets states that xH=yH if and only if x is an element of yH. Then, because c is an element of both aH and bH, cH=aH and cH=bH. Therefore, aH=bH. QED.

Lagrange's Theorem
Claim: The order of the group G is divisible by the order of the subgroup H; the number of distinct left cosets of H in G is |G|/|H|.
Pf: Let G be a finite group and let H be subgroup of G. Let a1H,..., anH be the distinct left cosets of H in G. Then, G=a1H&...&anH. However, by Property 4 of the Lemma of Cosets, these are disjoint cosets. Then, |G|=|a1H|+...+|anH|. However, |aiH|=|H|. Then, |G|=|a1H|+...+|anH|=n|H|. So, the order of G is divisible by the order of H. QED.

Corollary 2 of Lagrange's Theorem
Claim: The order of an element a of a group G divides the order of group G.
Pf: Let there be a group G with an element a. Then, 'a' is a subgroup of G. By Lagrange's Theorem, |'a'| divides |G|, but |'a'|=|a|. Then, |a| divides |G|. QED.

Corollary 3 of Lagrange's Theorem
Claim: Groups of prime order are cyclic.
Pf: Let G be a group of prime order with an element a such that a does not equal e. Then, n|'a'|=|G| by Lagrange's Theorem. However, because a does not equal e, |'a'| must not be 1. Because |G| is prime, the only other possibility for |'a'| is |G|. So, |'a'|=|G|. QED.

Corollary 4 of Lagrange's Theorem
Claim: a^|G|=e.
Pf: Let a be an element of a group G. Then, by Corollary 2 of Lagrange's Theorem, |a|k=|G|. So, a^|G|=a^(|a|k)=(a^|a|)^k=e^k=e. QED.

Ok, there are some other properties that MAY be on the exam. But I don't think I want to prove those ones. They're a pain. I almost guarantee that he'll ask for the proof of Lagrange's Theorem (and the requisite Property 4 of the Lemma of Cosets), and then he'll ask for one of the following: the Corollaries of Lagrange's Theorem, Commutativity of Disjoint Cycles, or some isomorphism properties. I've got it in the bag, kids. Ok, time to go review the definitions, then start making a reference sheet of groups -- that's going to be clutch.

Study Guide for Group Theory Midterm 1

This is my attempt to make a sexy study guide for my upcoming Group Theory Midterm. There'll be 3 parts: 1) Definitions, 2) Proofs, and 3) Properties of Groups Z(n), U(n), S(n), A(n), and D(n), with special focus on whether a given group is Abelian, cyclic, normal, etc.
This is the first section, Definitions.
I am expected to know the definitions of the following terms:
1: Even and odd permutations
2: Group isomorphism
3: Coset of H in G
4: External direct product
5: Normal subgroup.
So, here goes.
1: Even and odd permutations
A permutation is even if it is the product of an even number of 2-cycles and odd if it is the product of an odd number of 2-cycles. Specifically, if a permutation has an even number of elements, then it is an odd permutation, and if it has an odd number of elements, it is an even permutation.
2: Group isomorphism
An isomorphism P from a group G to a group G* is a one-to-one mapping from G onto G* that preserves the group operation. That is to say, an isomorphism is a function from one group to another and has three properties:
a. P is one-to-one; that is, every element g in G maps to a distinct element g* in G*.
b. P is onto; that is, every element g* G* is mapped to by a distinct element g in G*.
c. P is operation preserving (O.P.); that is, for all a,b in G, P(ab)=P(a)P(b).
Here are some notable properties of isomorphisms acting on elements and groups:
a. P maps the identity e in G to e in G*.
b. For every integer n and for every element a in G, P(a^n)=P(a)^n.
c. G is Abelian iff G* is Abelian.
d. G is cyclic iff G* is cyclic.
e. |a|=|P(a)|. Then, if G is finite, then G and G* have exactly the same number of elements of each order.
f. If K is a subgroup of G, then P(K) is a subgroup of G*.
3: Coset of H in G
Let G be a group and H be a subgroup of G. Then, for any a in G, the set {ah| h is in H} is denoted aH and is called the left coset of H in G containing a. A similar definition exists for the right coset.
Here are some properties of cosets:
a. a is in aH. Always, kids. Always.
b. aH=H iff a is in H (note that this makes aH a subgroup G, since H is a subgroup of G; this is the ONLY time that aH is a subgroup of G).
c. aH=bH iff a is in bH (or, similarly, iff (a^-1)b is in H).
d. EITHER aH=bH OR aH&bH={0}. Super important to know this. Super crucial.
e. aH=Ha iff H=aH(a^-1).
4: External direct product
Let G1, G2,..., Gn be a finite collection of groups. The external direct product of
G1, G2,..., Gn is denoted for this blog as G1#G2#...#Gn is the set of all n-tuples for which the ith component is an element of Gi and the operation is componentwise.
What does all that mean? It means that the external direct product is the set of n-tuples (like an ordered pair, but an ordered... n-tuple) where the first element is produced by using the operation of G1 on all of the "first" elements, the second element is produced by using the operation of G2 on all of the "second" elements, etc. This is weird. I don't quite get it. I'll return to this.
A few things to know about external direct products:
a. The order of an element in a direct product of a finate number of finite groups is the lcm of the orders of the components of the elements.
b. If G, H are cyclic, then G#H is cyclic iff |G| and |H| are relatively prime.
5. Normal subgroups
A subgroup H of a group G is called a normal subgroup of G if aH=Ha for all a in G. This is denoted by H|>G, for this blog.
The normal subgroup test:
A subgroup H of G is normal in G iff xH(x^-1) <= H for all x in G.
Ok, I'm tired, so I'm going to read myself to sleep now. A couple of proofs tomorrow, probably. Probably Disjoint Cycles Commute and the Four Corollaries to Lagrange's Theorem. Fun stuff.