Group

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Group
. k/ D- |7 [) K. H6 z% K7 Q: L' GA group is defined as a finite or infinite set of Operands
0 {/ G: z6 W' C+ | (called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
4 a" m4 Z# @5 U9 A$ I to form well-defined products and which furthermore satisfy the following conditions:
' x1 J2 ^7 v, W0 F$ O1. Closure: If and are two elements in , then the product is also in .
2. Associativity: The defined multiplication is associative, i.e., for all , .
: k, i+ V" q% a: e' c& g% }9 ~3. Identity: There is an Identity Element
* P5 ^3 N+ _( ?1 r4 K8 [& o4 l (a.k.a. , , or ) such that for every element .
0 U9 i- R7 l" Q' v$ ?$ x4. 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
4 T0 k& [! K" z7 Y- A7 O for which every element is invertible. A group must contain at least one element.
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, {1 Q+ C) Y# q& |The study of groups is known as Group Theory
: v, {2 F! H; T, c9 j. If there are a finite number of elements, the group is called a Finite Group
and the number of elements is called the Order
of the group.
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& I  L5 J# d/ W/ ]2 [Since each element , , , ..., , and is a member of the group, group property 1 requires that the product
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: f1 M; Q# G" L: Q6 z4 `must also be a member. Now apply to ,

(2)

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so
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0 A/ v* g7 V, c) M; twhich means that
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OLS

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