Group

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Group
: u4 P/ k" i. g- P' X  T2 NA group is defined as a finite or infinite set of Operands
( B# T2 v, [( F" ] (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 .
3 K  I* F- a4 Y8 s2. Associativity: The defined multiplication is associative, i.e., for all , .
+ _  g4 |- K. ^3. Identity: There is an Identity Element
: C* ?) b# d2 N1 O, A8 Q2 b4 ^0 J# g (a.k.a. , , or ) such that for every element .
. Q* }* F# ?% k1 g$ ^: w. N4. 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
1 s. K1 ~7 y. ?7 V. n3 ]/ d" g% r. If there are a finite number of elements, the group is called a Finite Group
7 z3 v0 V3 `- S7 } and the number of elements is called the Order
. Q1 I( r& \5 W( d+ p* \ of the group.
1 }8 I  {. O4 k; j$ s. v6 p
Since each element , , , ..., , and is a member of the group, group property 1 requires that the product
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(1)

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: v; X+ W# u+ d4 q. ?: Zmust also be a member. Now apply to ,
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2 m& ?1 O# B8 {

(2)

But

8 o# v- ?! D2 `5 g& l
4 e: I  u; t6 @; t5 [$ \
1 C4 W6 L1 W) g" B# D( y	(3) " a3 w8 D+ x# t7 P" O$ z/ B+ v& t, A8 A

so

# x* l  V& w/ l

(4) 9 z& X6 u3 k6 l( ?8 o

+ h* q/ u/ b$ b* t9 T1 Y

/ R3 h$ z+ w4 j/ c9 x! K* b! Dwhich means that
/ _7 ]. U$ R) s( b# r/ _

/ x& t9 O$ y  x. K1 B

(5)

and
0 |' \! b# M6 W0 G

A1 Q6 v& b) j1 T

(6) 1 ?4 _  r+ ?2 y

; h( J- p: E) F* n5 J# H

! Y0 u9 e7 }8 Q8 Z. G9 o
8 M3 K: T" o* b2 ~& K7 }

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