Group

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Group
A group is defined as a finite or infinite set of Operands
/ u4 p5 l4 _' P- ?9 u (called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
% H+ X, l3 C& L4 T5 i to form well-defined products and which furthermore satisfy the following conditions:
8 E, O0 j( d- E: j1. Closure: If and are two elements in , then the product is also in .
' K; m2 D6 p4 P( \+ N2. Associativity: The defined multiplication is associative, i.e., for all , .
3. Identity: There is an Identity Element
+ p9 \' @( a& w6 h3 r3 c  k (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.
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, v+ }/ J) d& E; K- S5 P, g2 hThe study of groups is known as Group Theory
1 H, u0 @) ^+ H1 R5 j0 P. If there are a finite number of elements, the group is called a Finite Group
( R: n9 h) S% b/ E2 g and the number of elements is called the Order
( F" X5 r8 {! b; W1 w! `. m of the group.
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Since each element , , , ..., , and is a member of the group, group property 1 requires that the product
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9 i" Y' D9 }( E7 jmust also be a member. Now apply to ,

(2)

+ ^/ w/ B2 K5 K4 Z3 e9 o2 a
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3 }* J; a# ^0 K; G5 P0 SBut
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	(3)

so

8 S9 w6 S5 Y9 H# i' {: a

(4)

2 F, ~1 C: X. Z# Y. `which means that
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(5)

" j, b, A, k- h5 O4 Q; L. }* L1 Dand
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OLS

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