Group

OLS

Group
, Z$ }$ K- ~1 u6 E/ QA group is defined as a finite or infinite set of Operands
(called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
, Q$ ~  l+ i/ d( U to form well-defined products and which furthermore satisfy the following conditions:
4 i' _1 D) I2 @6 e! Y* C1. Closure: If and are two elements in , then the product is also in .
: z& R# _0 c: n- M2. Associativity: The defined multiplication is associative, i.e., for all , .
7 J! e3 x/ j% R/ u" Q& ^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 .
2 I% s) v* \  o1 tA group is therefore a Monoid
: u. B, V: I1 J, e& K/ O5 B for which every element is invertible. A group must contain at least one element.

( j' n4 S/ P% l7 MThe study of groups is known as Group Theory
. If there are a finite number of elements, the group is called a Finite Group
" r. _9 U7 ]) v7 N9 R5 x! p4 e& \; F; I and the number of elements is called the Order
of the group.

Since each element , , , ..., , and is a member of the group, group property 1 requires that the product

(1)

) D6 R( B3 m$ y' i- k! ~( _0 @

+ k+ _9 x7 \2 _must also be a member. Now apply to ,
2 d8 }6 g) F; u# `/ x4 q
  p# P1 [5 z8 F7 |

(2)

' C- d6 y% |( @1 S  B9 D
- _. z! \' d, D+ v7 H. ^1 PBut

# L0 h0 S0 W% D6 @6 D) B
( P! B0 G3 y" e0 q	, x& ?; j9 A5 B+ Q' `) `* ?$ _( D
	(3)

so
8 J' M$ d: }& Z$ T* _: d

(4)

which means that
7 Q3 e! L0 q; p7 o) L$ \* F

(5) 8 q0 B4 H6 r: Y# u

9 c3 _# Q" B+ c9 {9 G. _, @

5 G: L& u8 L+ g7 n: uand
5 F* d( _3 Y3 z8 d8 D# V$ S% @

6 x; o: |" ^5 _3 M

(6) ! J5 |3 _4 n3 @3 L. @- [

2 G% L9 ]5 P+ m2 |4 H( B' q

/ a" j6 ?4 U7 f$ u  i2 N
! s$ z3 T  m6 _; H% b. I

OLS

这个帖子乱。