Group

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Group
A group is defined as a finite or infinite set of Operands
3 O/ E: A* L+ q. L# W0 \0 f9 } (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 .
2. Associativity: The defined multiplication is associative, i.e., for all , .
: w) w' r7 N3 q' g3. Identity: There is an Identity Element
(a.k.a. , , or ) such that for every element .
8 y6 E! [! C3 _$ d  R1 x4. 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
for which every element is invertible. A group must contain at least one element.

The study of groups is known as Group Theory
. If there are a finite number of elements, the group is called a Finite Group
and the number of elements is called the Order
+ w6 Y: K. a, U  ~: B+ b of the group.
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Since each element , , , ..., , and is a member of the group, group property 1 requires that the product
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(1)

must also be a member. Now apply to ,
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(2)

; t7 ?( @' c: b7 h5 J1 C! [4 B4 MBut

	* a3 }) j& G7 W
2 R5 G4 S6 U$ p9 @# m	( x  o" e& |! y: D4 Y
3 `3 R+ S% C( ]6 {* L5 Y	(3) $ y& p; `4 {1 W1 |

so

' U8 R+ T  t' l! Z3 b% E

(4) / F+ D. [% F0 i/ [6 x2 S

" ~/ p# H% f7 p. B7 S3 [- Y% i
; m8 D$ M8 M9 u3 }& Ewhich means that
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& s5 q1 s( H0 U0 b9 @4 [+ \2 N. W

(5)

/ F! k! O0 k9 D. M
+ _- X& u3 n  h$ g0 `' }
and
5 g8 v0 U& x* v5 K

(6) , W5 _/ A/ L/ X  E7 `9 `

" X3 l  K' K0 O1 n. Q. f7 e
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OLS

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