Group

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Group
A group is defined as a finite or infinite set of Operands
  |* A! V0 o4 Q) [% M: k, F (called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
1 P5 k2 x$ x6 \7 |$ `9 h% z) m 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
% K6 F8 I4 Y# v& Y (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 .
6 g3 Y& m. u' }" o8 K5 y, Q2 dA group is therefore a Monoid
* e; c$ k) V( N0 V3 v, F: x for which every element is invertible. A group must contain at least one element.

The study of groups is known as Group Theory
- ?8 P/ u8 |* y# w+ J. k/ S. If there are a finite number of elements, the group is called a Finite Group
% f. Q/ {# v+ P* V and the number of elements is called the Order
+ a, L, y9 Q* X% B! i5 x of the group.

Since each element , , , ..., , and is a member of the group, group property 1 requires that the product

(1) . e" O' J8 _: P: K4 h& g# @5 M

% I( W6 q, H, \' K. Vmust also be a member. Now apply to ,
6 E  k) X. e6 h. K0 [

6 x4 J2 K2 y5 b' {$ A9 Z

(2) 6 x8 c6 h$ v2 I* v) c/ U! X9 D

+ X5 E. G7 |6 P2 W8 e
7 B2 c; c& R4 R- q
4 l5 I# G0 \2 t: b, c% |But

. w$ r  |4 b( @" J
. m" _, v; k' u! U, P	: p2 k+ G* i7 \
	(3)

so
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(4) $ @9 J7 d/ M4 U0 B

+ E8 r: l, h) t3 Dwhich means that

$ u3 \* |! F. o4 w+ f

(5)

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" @: h$ E4 ?& L9 Hand

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(6) ; @6 _  A  ~5 s1 f! C* k

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