Group

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Group
A group is defined as a finite or infinite set of Operands
6 [: x$ @& F6 f. W) ?2 a (called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
, _+ @" {" N2 g# r: O$ G9 J to form well-defined products and which furthermore satisfy the following conditions:
& ?9 T7 f6 R8 c3 I1. Closure: If and are two elements in , then the product is also in .
" S9 K2 |8 Y* V- |. q: |; J2 v2. Associativity: The defined multiplication is associative, i.e., for all , .
) {1 Y' S! q! P! n; {  P3. Identity: There is an Identity Element
(a.k.a. , , or ) such that for every element .
: c: K+ S, }& I, W7 M" p4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of .
A group is therefore a Monoid
: [# [: G3 h' J* s" R for which every element is invertible. A group must contain at least one element.

The study of groups is known as Group Theory
$ r* W& Q9 X2 ^( N- ^* C" T  M: I. If there are a finite number of elements, the group is called a Finite Group
3 U/ G: I# r1 Q  n and the number of elements is called the Order
. U! E/ t# U- c) J of the group.
) y1 m7 [9 L# ~2 f
. Z+ i* Z! J, k& z9 H7 d$ zSince each element , , , ..., , and is a member of the group, group property 1 requires that the product

(1)

must also be a member. Now apply to ,
) L3 {5 b0 I. O3 a

(2) - }( @. J6 A: j' y/ R* c. c

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  |" A4 C4 H8 ^& J, t$ e' e6 c' G
But
, B0 i/ Y; k2 M+ H9 F

$ C8 l8 S5 b6 ?  j	5 _9 D/ {: C% S
; p- f6 u7 r) V' M0 A1 I. v	(3)

so
- u- u  f8 p0 o9 k  Z9 U- d0 m  F; k9 G

(4)

7 U. J+ T2 d% m2 {

1 f- p+ X6 ^1 {* Vwhich means that

(5)

/ l$ V1 q: P9 I) M  X
) ~9 o0 ^2 A' r( M# g8 J
+ @; c+ D. ?" jand
' m3 N. ~9 e( J/ _  F" {+ r( X5 Q0 o

(6)

& ]/ E3 n% y* g, i

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