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
( F9 `. T+ Y( x  ?! d1 S7 ` to form well-defined products and which furthermore satisfy the following conditions:
# z( z+ }# M: Z1 G1. Closure: If and are two elements in , then the product is also in .
6 q! G# r& a  V) _& E' \1 ]2. 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 .
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
for which every element is invertible. A group must contain at least one element.

3 p7 j+ ^1 g0 r9 Q( u+ H8 a3 HThe study of groups is known as Group Theory
% ]0 B9 c, p5 \! u. If there are a finite number of elements, the group is called a Finite Group
6 Y, g" H; D$ T! l, ^5 }" z5 m  ~ and the number of elements is called the Order
of the group.
+ J) _6 ~* K; R
' i) r3 a5 r# eSince each element , , , ..., , and is a member of the group, group property 1 requires that the product

7 j6 ?8 o1 G3 |$ ^

(1)

; \1 Z5 [2 t$ }6 x" ~* S% T

must also be a member. Now apply to ,
1 O7 W( T. Y% _8 D5 U/ O; E$ S

( W( y$ D. z% Z; p

(2) 3 W. S$ L) A+ R, {0 M

: C. W0 j5 a5 u% P7 h- L7 g
But
' ]$ ~+ {. i2 _' R$ L8 v& x3 v

	% P+ n1 ~4 O8 h2 T/ V  x8 j
6 T4 q1 q, h; ~1 X* U
) g5 T- C2 P- n- k6 |9 ^	(3)

so

(4)

( D* z2 h7 g: ]which means that

8 ~' X! Q% h/ R

(5)

$ }6 y2 m; z5 ]: d" h) V" k9 ~: Qand

9 H1 Y- |6 {% [, D: M; d& e+ F

(6)

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