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
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 .
7 M( _+ T% w/ }2. Associativity: The defined multiplication is associative, i.e., for all , .
" I* o! P$ u8 X* t3. 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 .
A group is therefore a Monoid
" j& T3 }5 V( H* ?2 V for which every element is invertible. A group must contain at least one element.

The study of groups is known as Group Theory
. u( f( w' b% o: w- `- y. If there are a finite number of elements, the group is called a Finite Group
8 H/ x" T' l: s5 s$ ~6 V and the number of elements is called the Order
! T, x' |- f1 ^+ U 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) $ O5 X7 l4 i# e7 [' ~

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so

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(4)

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" Y" l$ \, q2 D- b: a, s( s9 b! zwhich means that
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(5)

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1 O" P6 {4 h2 O* G% g" q$ p8 z, @and

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(6) ; m5 A4 m& V9 l8 V0 x, f; S& U

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