Group

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Group
* a; b2 H  Z5 c# O# P8 YA group is defined as a finite or infinite set of Operands
# l* L3 M6 J. i( b0 J; u0 [$ F# |$ e (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 .
+ n' d# d6 d9 _) Z. M2. Associativity: The defined multiplication is associative, i.e., for all , .
! z( Q! x4 a" i3. Identity: There is an Identity Element
$ k, z" K" R4 ` (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 .
. X" o6 O, B0 \" C6 jA group is therefore a Monoid
for which every element is invertible. A group must contain at least one element.

The study of groups is known as Group Theory
8 e6 Y) {0 |/ t' Z. If there are a finite number of elements, the group is called a Finite Group
2 f9 [% z4 x% G6 \1 o, G and the number of elements is called the Order
3 J# d6 o/ e7 g of the group.
  P' E! \. e$ ]4 b
Since each element , , , ..., , and is a member of the group, group property 1 requires that the product
$ m" \4 e6 _( ]" u0 j

(1)

9 q- q3 [- ?9 ~4 v' r' kmust also be a member. Now apply to ,

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0 D7 ^6 Q1 d' B( N) _  ?: BBut
7 _) k9 I* A" c& i3 B+ v$ O2 S* I

	; z8 {4 ~1 M4 V! `
# [5 I2 a; N* l4 }	(3)

so
! I, `3 _! m5 |

(4)

) C' F8 f6 t$ g# G% G, l/ L
( g+ k6 [: _) Iwhich means that
0 c- b* S2 q/ t; W: |" I

! K% E2 I+ O, b+ W

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and

(6)

0 Z$ [5 |! W1 {% X9 Z" r; |

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