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MathematicalGrouP

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, (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.

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.

Sets of BiJections form groups under composition, providing they are closed under this operation and inverses.  A HomoMorphism from a group to a such a set is called a representation thereof, and it is called faithful if different group elements correspond to different BiJections.  CayleysTheorem states that every group has such a representation, so the study of groups can be considered as the study of permutations.

A subset of G closed under * forms a SubGroup in an obvious way.  Every group contains at least two subgroups, itself and the trivial group containing only the identity.  Every other element is contained in a minimal subgroup isomorphic either to the IntegerNumbers or some ModularArithmetic, treated as groups under addition.

The KerneL of a group HomoMorphism consists of those elements that get mapped to an identity.  As it turns out, the possible KerneLs of a group are precisely the NormalSubGroups - those subgroups H such that g*H=H*g for all g in G - and this fixes the homomorphic image of the group up to isomorphism.

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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.

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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.

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'''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.