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
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1. Closure: If and are two elements in , then the product is also in .
& ~$ O* P! |. i9 F) C( x2 v4 v2. Associativity: The defined multiplication is associative, i.e., for all , .
3. Identity: There is an Identity Element
(a.k.a. , , or ) such that for every element .
  l7 p8 I0 ]  X5 ^/ q0 s4. 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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The study of groups is known as Group Theory
. If there are a finite number of elements, the group is called a Finite Group
and the number of elements is called the Order
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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)

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must also be a member. Now apply to ,

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(2) 2 r8 Y  v8 G) l0 T) P+ z

2 y. i7 \0 ?# I4 \! d; ?9 H+ T/ N& Y

But
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	: ?, l5 T7 _# Z- d1 I
	(3)

so
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(4)

& ]* ^% \! B4 f! [" g+ T
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& p4 q# T, R9 i* P1 P! k  \which means that

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(5) 2 }* P7 l/ H9 k! [2 I; @& H6 J

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