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Given a Group (G,*) defined as:

G in a NonEmpty? SeT and * is a BinaryOperation?, such that:

First Theorem:

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

        a*e=e*a=e and a*e=e*a=a.

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.)

Notice the method of proof, which is the same for both theorems and quite common in MathematicS?. It is called, among other things, the Indirect Method of Proof and Proof by Contradiction.

Four More Elementary Group Theorems

I. For all a. b belonging to a GrouP (G,*), if a*b=e, then a=b^-1 and b=a^-1.

II. For all a,b belonging to a GrouP (G,*), (a*b)^-1=b^-1*a^-1. III. For all a belonging to a GrouP (G,*), (a^-1)^-1=a. IV. For all a,x,y, belonging to GrouP (G,*), if a*x=a*y, then x=y, and if x*a=y*a, then x=y.

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Last edited February 15, 2001 1:50 am by RoseParks (diff)