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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