Group

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Group
A group is defined as a finite or infinite set of Operands
! @. M6 p- B$ ~" _8 v$ B) r, D (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 , .
3. Identity: There is an Identity Element
7 H! G' z0 U8 k# q9 E (a.k.a. , , or ) such that for every element .
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 .
A group is therefore a Monoid
% P- x6 m5 a7 T# ^; ]. b0 L/ m 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
, B- n+ ^& x! I and the number of elements is called the Order
of the group.
5 B2 s$ k! n2 m: [
- F4 S0 L( B, X6 }2 b6 Q  ]Since each element , , , ..., , and is a member of the group, group property 1 requires that the product

* h, M" @  \0 \4 d8 S

(1) 4 k9 g7 o3 Z: P1 X8 Y, L' U

5 G* _9 i' g* e6 k: Z0 l. s. _; B
. I1 H* B$ R5 G9 @$ T
must also be a member. Now apply to ,
% ^$ A; |! x$ _9 ^" G- ~
( I' |. z+ q* V. s

(2) 0 A2 @+ L: R9 K! g! W

+ [' y% D7 v, H) k/ S/ |9 D1 l2 v
But
& o7 R+ T% Z& }" P" z

	l% E! M9 Z4 z0 F  t0 j4 ]( t
* e" z6 s3 S/ L" a. l. d/ a6 Y
	(3)

so

(4) / Z5 x# ^# S9 k" H3 B. [

2 X9 u0 _  u. X" G! R; D. T

" L3 @+ n2 ^- m- ywhich means that
! V) P  d: @1 V. n! p3 I& n( p& ~/ T

5 j' A8 k1 x2 A% l$ a5 ?

(5) * w7 V; H/ ~* U

0 a/ U4 u; p. x0 J6 q$ j
, P; Y3 n( K( `! Z! F7 D
- [' R3 e* p- E5 n4 g! Cand
& u! H% g& U$ y

(6) 4 _% L& ?1 l. A% n7 C, W! ^

& \9 P, a) Z' |1 k/ z; b  F$ W
4 D  L2 D: `$ D' r: j  d  R1 e
" @& w6 k1 U7 r# V8 Q$ P
) I. P# I" A) {3 o, r% r
2 a1 {) m- u3 R+ p: \

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