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
to form well-defined products and which furthermore satisfy the following conditions:
8 F' b: ^) d7 m  l$ p1. Closure: If and are two elements in , then the product is also in .
; e6 U# o+ Y' a; H7 d; v2. Associativity: The defined multiplication is associative, i.e., for all , .
3. Identity: There is an Identity Element
* x8 h) v! c+ j0 ~4 n+ A (a.k.a. , , or ) such that for every element .
3 G4 z4 {5 ?# V# k5 E2 ], [* [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# r. R2 M6 Q- C; m  ~+ EA group is therefore a Monoid
2 I- m  u* v  V" C for which every element is invertible. A group must contain at least one element.

. v+ T0 \4 `( A( v, y) b) eThe study of groups is known as Group Theory
& C/ w3 t  d7 l8 y. If there are a finite number of elements, the group is called a Finite Group
' P# Q# Q- t, Y$ [& y, g0 S 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

$ t0 \2 ]6 b( @+ E" `

(1)

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must also be a member. Now apply to ,

(2)

But
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	8 ]* ~! |2 x. ]3 \+ m
	(3) 9 b' W4 e% J7 O7 H& }+ I* Z1 {

so

(4)

which means that
8 U1 Z  r1 ]5 L$ E1 p6 t8 g

- H  y. k2 `8 X3 h

(5)

0 C2 `7 u5 y9 i- [
and

2 I# X( Z0 N1 d* ~0 a/ d& ~

(6)

  e( o# w1 \4 N( p7 M
, i  P# E& h" \- s; s/ h2 {

2 x) Z+ s' W) e) v3 L( H
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