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# ElementaryGroupTheory

```FirstTheorems

Given a Group (G,*) defined as:

G in a NonEmpty SeT and “*” is a BinaryOperation, 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*e=a.
*4).  Every element in (G,*) has an inverse, that is, of a belongs to (G,*), there is an element b in (G,*) such that a*b=b*a=e.

First Theorem:

The identity element of a GrouP (G,*) is unique.

*Proof:
*	Suppose there were two elements, e and e’ (G,*), such that for all a in (G,*)
a*e=e*a=e and a*e’=e’*a=a.
*	Then e*e’=e’ by the definition of identity element (3).
*	And e*e’=e, also by the definition of the identity element (3).
*	Then e=e’.
*	The identity element in a GrouP (G,*) is unique.

Second Theorem:

Given a group (G,*), and an element x in (G,*), there is only one element y such that y*x=x*y=e. ( The inverse of each element in (G,*) is unique.)

*Proof
*	Suppose there are two elements, y and y’ in (G,*) such that for x, an element of    (G,*), y*x=x*y=e and y’*x=x*y’=e.
*	Then y’=y’*e, by definition of identity element (3).
*	Then y’=y’*(x*y), since x*y=e.
*	Then y’=(y’*x)*y, by the associativity property of (G,*) (2).
*	Then y’=e*y, since y’*x=e
*	Then y’=y, since e is the identity element (3).
*	Therefore, the inverse of an element x in a GrouP, (G,*) is unique.

Notice the method of proof, which is the same for both theorems and quite common in
MathematicS. It is called, the Indirect Method of Proof.
*	First, one assumes that the proposition one is trying to prove if false.
*	Then, one tries to get a contradiction.
*	If this is successful, then the assumption that the proposition is false, is, itself, false. Hence, the proposition is true.

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