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,*), 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.

The proof of the '''uniqueness of the identity element''' and the '''uniqueness of the inverse of an element''' in a GrouP, falls under ElementaryGroupTheory.


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.  We usually only use additive notation when the group is commutative, ie a+b=b+a for all elements.

Some important concepts: SubGroup''s'', PermutationGroup''s''.  Some examples of groups with additional structure are RinG''s'', ModulE''s'' and LieGroup''s''.
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Basic examples of Groups

+Signed integers
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.


+Group of geometric transformations of the plane.
An important source of groups are geometric transformations of various objects in various spaces. As a simple example, let's take the set of all translations (moves) on the plane. The operation is the composition, noted here o (the composition of two operations consists in applying the two transformations one after the other)

1)	If a and b are translations of vectors A and B, then a o b is also a translation of vector A+B

2)	Composition of translations is associative as is the additions of vectors. A+(B+C) = (A+B)+C.

3)	The identity element for this group is the translation of vector zero.

4)	The inverse of a translation of vector A is the translation along the opposite vector (same length and direction, reversed orientation).

One can build more generally the group of all distance-preserving transformations of a space or set of points. In this case, the proof of the group axioms is bit more complicated and interesting. (OpRgAg). 
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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 RinG''s'' and FielD''s''.