Group

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Group
6 ?: r: b' m9 QA group is defined as a finite or infinite set of Operands
(called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
- Q5 Q: L) e, H: e: M to form well-defined products and which furthermore satisfy the following conditions:
+ [2 w" Z& v( J" b8 s; J$ M1. Closure: If and are two elements in , then the product is also in .
! u) t; d5 M$ J4 p, ^2. Associativity: The defined multiplication is associative, i.e., for all , .
8 D# G/ }2 q' S" v0 i' ^8 R# p3. Identity: There is an Identity Element
(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 .
5 S; y9 |4 J0 ]' `$ m4 c2 lA group is therefore a Monoid
for which every element is invertible. A group must contain at least one element.
- w# e3 w9 K" y
  u! k4 Z" J! r5 VThe study of groups is known as Group Theory
6 y5 p, f$ M  G  Q' B. If there are a finite number of elements, the group is called a Finite Group
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
; Q8 L0 s6 R# d! ^; [4 h, z

(1) 5 C* a* q" W. L; g

must also be a member. Now apply to ,

(2) # N3 n# f7 {) d$ i& z+ ^6 F  p$ B

0 i, e4 }# [3 x. a% h
9 v; K, |' W0 Y1 Z# @( Z* l' E; L) t
# S; T- z  P2 k0 L% [+ S% ]But

, b, U. {" N' P9 z
	(3)

so

(4) " g! k6 U, K- \& q

0 N! A! K: `. R5 Y% R

which means that
- L5 D  `3 V  Y

7 I: m2 c' @! H# S/ P/ h

(5) & g3 o* z' E6 \! D0 V2 Z5 E

and
3 C: m- p' k: N7 z7 ]5 }$ x

2 Y: l4 |- U: z1 _

(6) ( I: d& j' M0 {

$ v3 D/ M9 C  N9 a) z) C
& W; j6 W, u9 n0 \$ \
7 L( D9 }! k3 {0 \: p

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