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
5 a$ h/ J2 i: m& h0 d! d6 V, r to form well-defined products and which furthermore satisfy the following conditions:
: e' ^* f: V. [1 u. J" }0 E* F( q# H1. Closure: If and are two elements in , then the product is also in .
, R2 M4 \7 x. `; e1 f- s& \1 z( R2. Associativity: The defined multiplication is associative, i.e., for all , .
5 \; m) ^+ p% D2 N3. Identity: There is an Identity Element
1 R" y  I+ M/ H: Y, Q( g; R# h5 q$ N (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 .
# z! v% g6 ?/ U4 OA group is therefore a Monoid
7 o; H; L& \0 }- K) K, f for which every element is invertible. A group must contain at least one element.
; N4 W3 N& B) M) U/ l
The study of groups is known as Group Theory
. If there are a finite number of elements, the group is called a Finite Group
9 T9 b. c" k2 w$ y! V and the number of elements is called the Order
of the group.

Since each element , , , ..., , and is a member of the group, group property 1 requires that the product
; @; ~$ _* I: s) f$ {6 G

( `5 B0 Q, s3 p1 a! ^/ |

(1) & c( F; E3 K+ n% m, T1 s& ^

; L5 p, z9 R" f) M

& N. Q! S0 L/ }4 [1 Kmust also be a member. Now apply to ,

% `/ j+ m6 c* b" y* G

(2)

But

2 a1 n0 v. V  S5 W! b1 p	) ^) d5 o7 B; b, s- h( P+ Z; ?$ N
	, b$ p$ e% }/ f( M
: N7 K/ ?% B" W* h' \	(3) , s1 A: R8 w6 F. j# Q1 S

so

(4) 8 ^( w- h( D2 N5 l' J6 y

2 Z9 ^6 I  q9 ]: Q# K
( S# d+ n! o. v6 I  B7 qwhich means that
* Y9 K( N7 j5 Y1 c0 t6 z; W

(5) ! c. o5 D  \! U" t2 m! w& w

/ l0 U+ r5 T$ X

! t- L0 Z8 `; g# o2 Qand

& t: }* ~8 \, _& j* \& ]

(6)

0 Q) r- _' C' a; i5 I- e+ |* i


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