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'''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''.
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.


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 of a set of points in a space. In this case, the proof of the group axioms is a bit more interesting. (OprgaG). 

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


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?

'''Answer''':  No.  The rational number 0 is in Q, but does not have a multiplicative inverse.  If we add modify the constraint such that neither a or b may be 0 (i.e. the SeT is Q excluding 0, denoted Q\{0}) then (Q\{0},*) do form a group.  The inverse of a/b is b/a, and the remainder of the group axioms are simple to check.

(Q,*) does form a SemiGroup (i.e. a group lacking inverses)

Groups are important as a fundamental algebraic structure and as a basic tool for embedding the mathematical properties of objects, even properties of groups themselves because one can built groups out of existing ones (see GroupProductS, NoetherIsomorphismTheoremS, WreathProduct, BrauerGroup, DerivedGroup, GroupCenter, GroupCharacter, FittingSubgroup, CohomologyGroups). We can investigate groups with global properties like CommutativeGroups and NonCommutativeGroups, SimpleGroups and SolvableGroups, FinitelyGeneratedGroups, AlgebraicGroups, study how groups can be related via MorphisM''s'' and also build other structures based on the notion of a group, but with more than one operation and/or more properties. This brings us to RinG''s'', FielD''s'', ModulE''s''.