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
% b% w4 q4 `- P( U" {* ?0 ^( Q to form well-defined products and which furthermore satisfy the following conditions:
( ^) {- E9 Z  A0 y# B3 s4 T' g0 |1. Closure: If and are two elements in , then the product is also in .
# ^" l) \7 N9 l3 t; \; {: w2. Associativity: The defined multiplication is associative, i.e., for all , .
5 _6 E9 n* Y/ `; b8 _$ m+ d3. Identity: There is an Identity Element
4 a. D! X2 V: P% ^; X2 i2 M( b; C (a.k.a. , , or ) such that for every element .
- y( A7 L7 @$ A0 W4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of .
6 O1 ?# y" B  V9 O0 qA group is therefore a Monoid
4 ?  ^! M( v. ^2 w+ ? for which every element is invertible. A group must contain at least one element.
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The study of groups is known as Group Theory
/ Z6 S+ q7 A( h1 F. If there are a finite number of elements, the group is called a Finite Group
and the number of elements is called the Order
! ?4 T" T: i. n/ j$ W of the group.
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Since each element , , , ..., , and is a member of the group, group property 1 requires that the product
6 ~, @( s$ \% o) T0 m9 [

(1)

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, C' G5 f+ i* v% h- G. Z# I; D. emust also be a member. Now apply to ,
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) B5 H. M. M% V5 v

(2) 0 K* j! ^0 y: Y$ l

+ Y/ ^. H; S4 W, }5 Q1 `6 Q% a# r

But

	* ?9 l4 L  e- r
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	(3)

so

(4)

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which means that

0 }5 {! N5 C: R

(5)

5 N3 @, p. R. i8 c' @* B( s6 a

and

(6) & g- T( T, b' s* l$ B* g

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