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Standard Math Facts

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Various mathematical terms or facts will be defined, stated or proved.  This material is generally available in most abstract algebra texts (such as Lang's Algebra or Serre's Cours d'Arithmétique.)  This collection of definitions and theorems may be expanded depending on feedback.

 

A bit of number theory

Natural numbers are the positive "counting" numbers {1,2,3,4,...}. 

Given two natural numbers a and b, their greatest common divisor is denoted (a,b).  a and b are relatively prime if their greatest common divisor is 1.

The Euler totient function of a positive integer n, ø(n), is the number of positive integers less than n which are relatively prime to n.  [The symbol ø is not quite right, but it does display in html.]

If p is prime and r>0, ø(pr) = pr-1 (p-1).     

If (a,b) = 1, then  ø(ab) = ø(a)ø(b). 

 

 

Properties of Finite Groups:

  1. A group G has an operation (written additively or multiplicatively) mapping GxG to G.   There is an identity element e, an inverse for each element in the group and the operation is associative.  If the operation is commutative, the group is called abelian.
  2. The order of a group is the number of elements in the group.
  3. A subgroup is a subset of a group which is a group under the inherited operation.   (For example, integers with addition are a group and the even integers are a subgroup -- with addition.)
  4. In a finite group, the order of a subgroup divides the order of a group.
  5. A cyclic group is generated by a single element.   
  6. If H is a group of order n with the property that for d dividing n the equation xd = 1 has at most d solutions, then H is cyclic.
  7.  
  1. Field Properties
  1.  
  2.  
  3.  
  1. Finite Fields  
    1.    

 

 

 

Groups

Group homomorphism: a map from one group to another which passes the operation.

Let G, H be groups. A mapping h:G-->H is a group homomorphism if for al s,t in G we have f(st) = f(s) f(t). If e is the identity of G, then f(e) is the identity in H. Similarly, f(s-1) - f(s)-1 (inverses). Thus f(G), the image of G in H is a subgroup of H. The kernal of f, or ker(f) is all elements x of G with f(x) = identity in H.

A group isomorphism is a homomorphism which is one-to-one and onto.

A cyclic group is generated by powers (or multiples if written additively) of a single element. If g is a generator, <g> = {...,g-2,g-1,1,g,g2,...}. If there is a number n such that gn = 1, then this is a finite cyclic group. <g> is isomorphic to either Z or Z/nZ, written additively, with 1<-->g.

Fact: If H is a finite group with the property that for any d|n there are at most d elements of H with xd = 1 then H is cyclic.

 

We will not need this but every finite abelian group is a direct sum of cyclic groups.

In non-abelian groups, the structure can be quite complicated.

 

 

 

Fields

A field (F,+,x) is a set with two operations, commonly written as addition(+) and multiplication(x). F is an abelian group under addition (identity usually called 0). .Fx = F -{0} is an abelian group uder multiplication. In addition, the distributive rule connects the two operations: a(b+c) = ab + ac for all a,b,c in F.

 

 

Finite Fields

 

 

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Last modified: July 21, 2001