Group

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Group
A group is defined as a finite or infinite set of Operands
(called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
to form well-defined products and which furthermore satisfy the following conditions:
1. Closure: If and are two elements in , then the product is also in .
% c: U; H' N8 \+ k2. Associativity: The defined multiplication is associative, i.e., for all , .
3. Identity: There is an Identity Element
(a.k.a. , , or ) such that for every element .
6 h" R8 u& J6 |  q4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of .
! Z0 \- }$ Y3 p9 L+ U" i& NA group is therefore a Monoid
for which every element is invertible. A group must contain at least one element.
! G; z8 n; J: R, F0 Y/ O0 o
) X- b7 j) y! d" UThe study of groups is known as Group Theory
. If there are a finite number of elements, the group is called a Finite Group
; H3 A# e, x% x5 d& B2 j and the number of elements is called the Order
of the group.
7 w4 t( z+ S% G4 Q' F0 ?
( F3 |: h( s9 nSince each element , , , ..., , and is a member of the group, group property 1 requires that the product

* y. N3 t1 Z4 L0 s

(1) : |4 [" u7 @4 f6 n' l0 V

must also be a member. Now apply to ,
5 z, |4 R2 S4 |6 `( a: b

1 N( m5 ?- K8 l1 }: ]

(2) + `; e+ d) s9 y5 ^; c

+ q7 M0 l( F  k$ i
6 q" Q# K- ]3 }  J
But
" I. @- X' g* {5 i) B

7 ?5 [1 t* k7 G# u" z5 U	, }8 e1 i. P. e, F/ S/ y
1 P4 N) N$ l+ \8 Y0 d	(3)

so
( o" O4 V" s( O4 ?

0 ~( d, Q/ [# _

(4)

5 Z$ d; M: ~( X" R+ K

& \" h* P' u  jwhich means that

(5) 0 y  {+ R: ]# e# V4 e4 T

/ X; u- @0 H# K. e" v
and
- Y' @7 j5 c2 @) o

(6) ; V  t8 R( v8 u2 ^6 {

1 x1 Z/ m  E6 O

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