Group

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Group
& c0 ?0 A* Q% z/ q1 pA group is defined as a finite or infinite set of Operands
7 Y: _; i. x. s  i1 z5 e. c! j! ` (called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
8 o* \9 l. P4 o 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 , .
% p8 D) I* V, R; T" j3. 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 .
* h! E$ R, `( g/ U2 |0 FA group is therefore a Monoid
8 C1 [! E$ r/ g/ D. s for which every element is invertible. A group must contain at least one element.
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The study of groups is known as Group Theory
& A/ t: L+ @4 m. W- m7 I. If there are a finite number of elements, the group is called a Finite Group
9 U4 z$ R1 Q& A and the number of elements is called the Order
# y2 @! N4 r/ c9 j9 K4 z 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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must also be a member. Now apply to ,

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(2)

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so

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(4)

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* o3 [- c: |, F% f# M- ^5 R, `which means that
1 U+ ^2 {+ W2 ?2 o

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(5)

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and
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(6)

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