Group

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Group
9 |1 |" l5 G  ]9 p- k! uA group is defined as a finite or infinite set of Operands
(called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
0 g/ Z  m8 t% Y2 ~' O5 ` 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 Y; j: {: m, v: j$ D2. 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 .
/ M) l% ^# n% R. x( U4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of .
! o: P1 P+ K0 Y; S* RA group is therefore a Monoid
for which every element is invertible. A group must contain at least one element.

* v, W; _% i' ~9 M/ rThe 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.
7 p. i, K. R: R+ G3 \0 D
Since each element , , , ..., , and is a member of the group, group property 1 requires that the product

(1)

: @% _) Q( y- n
. N/ b0 v, v5 rmust also be a member. Now apply to ,
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4 J1 o- [4 H6 o; R, P

(2)

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7 J5 }& g, K! d
5 g( _3 N0 g: C4 `% f* fBut
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2 S7 F: @/ \5 I- T9 x	(3)

so
$ Q, L/ A; {; n$ M8 Y' L% O0 O

(4)

: m: b/ A' {  o6 A5 ~0 Y. t$ k

: b  [5 S9 q" S+ `1 Zwhich means that
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8 k) w, V; G( y1 g6 l2 nand

(6)

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