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
! b2 P# l: ?1 k1 m6 `# Z0 ]& R# J6 y. e 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 .
/ p5 h& H8 s, a" R2. Associativity: The defined multiplication is associative, i.e., for all , .
3. Identity: There is an Identity Element
, r& v, E0 ?, L# J' v7 [ (a.k.a. , , or ) such that for every element .
7 f! [3 h9 l% J! E6 x" O4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of .
( U! ?0 B2 |7 d, n& E2 s7 UA group is therefore a Monoid
6 b" q6 \) ~- @4 S3 k# M 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
. If there are a finite number of elements, the group is called a Finite Group
$ s- s; q+ `! R' l, G2 {( v and the number of elements is called the Order
5 z% X1 W- {3 H4 H 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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(1) 1 B* l! h2 t* \: e: h" a$ l# {: L

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6 d. W  u% u) O! w4 U! z" Hmust also be a member. Now apply to ,

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(2) ! Y4 K3 Q8 G! S& o9 |

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But

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) s/ f! s% Q. }	(3)

so

5 l( {* `" _! e5 o7 c# M

(4)

which means that

(5)

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(6) 5 x4 b% \  C# D

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