Wikipedia 10K Redux by Reagle from Starling archive. Bugs abound!!!

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# MathematicalGrouP

```MathematicalGroup

The concept of a Group is one of the foundations of ModernAlgebra. Its definition 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 c 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.

The importance of groups lies in the fact that any set of permutations naturally becomes a group when you include all compositions and inverses.  In fact, any group can be represented in this way; this is called CayleysTheorem.

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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 c 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, b, and c 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?
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