Group

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Group
# ]# M1 R3 G; t' U! ~' EA group is defined as a finite or infinite set of Operands
(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
. Q5 G& X' f4 G (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
for which every element is invertible. A group must contain at least one element.
: J3 e9 x' x6 I) K; c8 `
$ q- o+ ?) [& pThe study of groups is known as Group Theory
. 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.

, ^- C5 p  T. E$ PSince each element , , , ..., , and is a member of the group, group property 1 requires that the product
! W, Z* k5 c% |6 a3 V6 M; F4 ~* v/ `

(1) ' B4 o+ E' m7 C

$ A! x2 O7 c+ L1 c

must also be a member. Now apply to ,
8 z( {; M  \/ g+ w) }5 x- F

+ Y  `% p0 O# H* |; n7 f) ]6 I, b

(2)

0 ], H* u2 y$ p, U' l- S
" }! ~9 o1 Z; ]6 X: y
But
4 S5 z3 \4 K# [

$ k) _! X6 P8 m0 c	W/ m+ }% E4 h( [
% d, V4 Z5 y1 |7 `# _/ `	(3) 3 k/ E. F2 y2 N5 {

so
7 }. Y; b+ r& Z' q* i2 N7 t

(4) 0 G* l2 {& t0 @

% C$ ]  Z7 w/ u4 C

5 G- X& ~4 L4 Z% E& owhich means that
% Y& v% x$ [# U0 K& x( _6 L+ T

(5) ( D" C0 z! l$ k/ A

, B! o; |2 |1 n* gand

0 ]/ o5 q1 c9 R" h

(6)

0 o3 k4 g  ~1 y5 q# ]) y
% Q; ~5 B8 `0 D1 I

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