Group

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Group
A group is defined as a finite or infinite set of Operands
* r2 T4 X* O* y8 l6 D3 E (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 .
) c5 f1 X) X2 u9 o2. Associativity: The defined multiplication is associative, i.e., for all , .
/ c, A# Z; `  m4 y2 k2 h( ~( S7 ~3. Identity: There is an Identity Element
5 f7 F% W% f3 U# Y (a.k.a. , , or ) such that for every element .
% ?- n7 o& V- d3 r+ ~8 Q6 }4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of .
( S. m  W$ k' `& V/ G# FA group is therefore a Monoid
! ~$ [( y& c$ W" _6 [7 R) [ for which every element is invertible. A group must contain at least one element.
* T1 O8 @. Y. E" V. o. F
- H/ x. `- L+ iThe study of groups is known as Group Theory
) l5 {- a; |( @  s. If there are a finite number of elements, the group is called a Finite Group
and the number of elements is called the Order
6 |) }" q/ I4 g; @9 N4 G of the group.

! Z% x2 _/ V1 f2 |8 TSince each element , , , ..., , and is a member of the group, group property 1 requires that the product

(1) & S2 |( \$ p/ N( \' o

, K  r# l" c1 d' w/ D; @must also be a member. Now apply to ,
6 _# t; v8 U$ R
+ t0 V1 q7 A' L9 ?$ R

(2) 8 {) q4 V$ A" E

But
6 y# v1 V; b( _- _/ E) X

	(3)

so
: v  k+ o. W5 o/ Z; g6 q1 S

7 a! }3 H- g8 w

(4) : a9 J( P: w8 h5 `# V* J

  ~3 e% C& X7 @0 ?2 z

- G9 ?) d1 C- e! n, M) Iwhich means that

(5)

$ j6 o! |$ N! `1 d; i: dand

(6)

6 _4 j" \( x/ y  ^, z1 @/ A( A

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