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MathematicalGrouP

'''MathematicalGrouP - definition'''

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.  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.

'''Subgroups, HomoMorphisms, etc'''

A subset H of a group (G,*) that itself forms a group under the restriction of * is called a subgroup.  Every group has at least two subgroups, G itself and the trivial group {e}.  Every other element is contained in a unique smallest subgroup, called a CyclicGroup, isomorphic to the IntegerNumbers or some ModularArithmetic.

A group HomoMorphism is a map from one group to another that maps products to products, and thence identities to identities and inverses to inverses.  If it is also OneToOne, it is called an IsoMorphism.  The structure of a group is determined completely by its IsoMorphism class.

The set of elements a HomoMorphism sends to e is called its KerNel, and is necessarily a subgroup H satisfying a*H=H*a for all a in G.  Such subgroups are called normal.  There is a homomorphism from G having any given normal subgroup as its kernel; the image of G is unique up to homorphism, and is called the FactorGroup G/H.

The intersection of any two subgroups H,K is a group, and if they are disjoint, the DirectProduct H*K is a group.  This is isomorphic to the Cartesian product of H and K under componentwise multiplication, which is something that can e taken even when H and K are not being looked at as subgroups.

A group without any normal subgroups is called simple.  The simple commutative groups are precisely the ModularArithmetic''''s of prime order; any other finite commutative group is isomorphic to a direct product of these.

'''Representations'''

The set of all BiJections on some set X form a group under composition.  This is called the SymmetricGroup on X.  The symmetries of mathematical objects are usually subgroups of the SymmetricGroup.  In fact any group is isomorphic to a subgroup of the SymmetricGroup on ''some'' set X; such a map is called a representation of the group.

The action of the group G defines a product GxX->X (action on the left) or XxG->X (action on the right).  Note every group acts on itself in an obvious way.  If no subset of X is closed under the group action, then the representation is called irreducible; other representations are arbitrary unions of reducible ones.
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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.