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MathematicalGrouP

MathematicalGrouP

The concept of a GrouP is one of the foundations of ModernAlgebra.


The definition of a GrouP is brief.

 
A GrouP is a non-empty set, say G and a binary operation, 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 b belong to G, (a*b)*c=a*


        (b*c).
3)	G contains an identity element, say i, that is, if a belongs to G then i*a = a and a*i = 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=i.

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A GrouP that we are introduced to in elementary school is the integers under addition. Thus, let I be the set of

 
integers={?-4,-3,-2,-1,0,1,2,3,4?} and let the symbol "+" indicate the operation of addition. Then, (I,+) is a GrouP.


Proof:


1)	If a and b are integers then a+b is an integer: Closure.


2)	If a, b, and b are integers, then (a+b)+c=a+(b+c). Associativity.


3)	0 is an integer and for any integer a, a+0=a.


       (I,+) 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 (I,+) has an inverse.

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Question: Given the set of integers, I, as above, and the operation multiplication,
denoted by "x" is (I,x) a GrouP?


1)	If a and b are integers then axb is an integer. Closure.


2)	If a and b are integers, then (axb)xc=ax(bxc). Associativity.


3)	1 is an integer and for any integer a, ax1=a.


       (I,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 (I,x) does not have an inverse.


For example, given the integer 4, there is no integer b such that 4xb=1.


Therefore, (I,x) is '''not''' a GrouP.

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'''Question''': Given the set of rational numbers Z, 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 (Z,x) a GrouP?