MathematicalGrouP

The concept of a Group is one of the foundations of ModernAlgebra. Its definition is brief.

A Group is a NonEmpty SeT, say G and a BinaryOperation, say, "*" denoted (G,*) such that:

1) (G,*) has CloSure, that is, if a and b belong to (G,*), then a*b belongs to (G,*).

2) The operation * is associative, that is, if a, b, and c belong to (G,*), then (a*b)*c=a*(b*c).

3) (G,*) contains an identity element, say e, that is, if a belongs to (G,*), then e*a = a and a*e = a. This turns out to be unique.

4) Every element in (G,*) has an inverse, that is, if a belongs to (G,*), there is an element b in (G*) such that a*b=e=b*a. This turns out to be unique.

Usually the operation, whatever it really is, is denoted like multiplication; we write a*b for the product of a and b, 1 for the identity and a^-1 for the inverse of a. Otherwise, the operation is denoted like addition; we write a+b for the sum of a and b, 0 for the identity and -a for the inverse of a.

Some important concepts: SubGroups, PermutationGroups. Some examples of groups with additional structure are RinGs and ModulEs.

A Group that we are introduced to in elementary school is the integers under addition. Thus, let Z be the set of integers={...,-4,-3,-2,-1,0,1,2,3,4,...} and let the symbol "+" indicate the operation of addition. Then, (Z,+) is a GrouP.

Proof:

1) If a and b are integers then a+b is an integer: Closure.

2) If a, b, and c are integers, then (a+b)+c=a+(b+c). Associativity.

3) 0 is an integer and for any integer a, a+0=a.

(Z,+) has an identity element.

4) If a is an integer, then there is an integer b= (-a), such that a+b=0.

Every element of (Z,+) has an inverse.

Question: Given the set of integers, Z, as above, and the operation multiplication,

denoted by "x" is (Z,x) a Group?

1) If a and b are integers then axb is an integer. Closure.

2) If a, b, and c are integers, then (axb)xc=ax(bxc). Associativity.

3) 1 is an integer and for any integer a, ax1=a.

(Z,x) has an identity element.

4) BUT, if a is an integer, there is not necessarily an integer b =1/a such that

(a)x(1/a)=1.

Then, every element of (Z,x) does not have an inverse.

For example, given the integer 4, there is no integer b such that 4xb=1.

Therefore, (Z,x) is not a Group. It is a weaker type of object sometimes called a SemiGroup.

Question: Given the set of rational numbers Q, that is the set of number a/b such that

a and b are integers, but b is not = to 0, and the operation multiplication, denoted by "x," is (Q,x) a GrouP?----

Groups are important, too, because they are a fundamental algebraic structure. We can investigate groups with added properties like CommutativeGroups and NonCommutativeGroups, and also build other structures based on the notion of a group, but with more than one operations and more properties. This brings us to RinGs, IdeaLs and FielDs.