Group

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Group
9 M5 \* _5 S6 LA group is defined as a finite or infinite set of Operands
+ d( B; O5 F) m (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 .
2. Associativity: The defined multiplication is associative, i.e., for all , .
9 B- ?- E5 n3 [/ o3. Identity: There is an Identity Element
(a.k.a. , , or ) such that for every element .
5 L  V  s2 X8 y# F4. 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
) {. g: k" j3 c( f3 `: C5 Y6 } for which every element is invertible. A group must contain at least one element.
8 K+ j2 Q1 S9 {( q3 |- ~
* M6 E6 H* @5 P) @8 R1 V$ uThe study of groups is known as Group Theory
  H( o( _/ s+ W  e. N1 P. If there are a finite number of elements, the group is called a Finite Group
and the number of elements is called the Order
& o( U' L% n# Z' ?- I of the group.
& _  Y. P; z9 T0 B
4 Z1 S( g" _& q& JSince each element , , , ..., , and is a member of the group, group property 1 requires that the product
; \( T9 v: r7 _2 j6 n( F$ m

(1) : E3 R" G5 u7 l8 e

* I8 H( ^; k& C( V: \7 Z/ amust also be a member. Now apply to ,
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(2) & [; A3 O- o) |: T8 ]4 Y

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2 ?; i9 Q0 R% }9 L* z' ]6 NBut

. Z1 B0 X  m3 ?+ i	. C& b" A2 w3 c, F- e5 A
( g; C# E! N! Y$ F/ K# B% z" j
	(3)

so

(4)

, }- u* z' t: S$ A

which means that
" P' b  T5 }+ K  H: O. X; h

(5)

and
, _6 D1 t* i% O4 K+ N

! R* C, X. k0 t& Z% m6 Q  {

(6)

$ L8 t$ O: P: k* \  [. L

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