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:
4 I# D% N, t; E7 s% g* D* i+ @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 , .
3 n; e7 `2 @  ^9 m' `3 B  b2 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
& f& t) g: {  i/ l. x/ ~* A5 g1 x for which every element is invertible. A group must contain at least one element.

+ ~2 L1 D, F1 r* t7 pThe 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
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) # v/ X" u) f; R' _

) t* P( Z) F7 c+ C% Fmust also be a member. Now apply to ,

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(2) 9 C7 R" |8 G$ n, }" N6 p

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6 P# Y0 x: d1 uBut

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so
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(4) , }8 i# u- ~9 Y7 L: v

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which means that
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(5)

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( j3 |/ p; G: {: b2 C/ E" ]and
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(6)

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