数学建模社区-数学中国

标题: Group [打印本页]
作者: OLS    时间: 2009-2-2 22:25
标题: Group
Group
+ `- y- [% B4 i  A2 {A group is defined as a finite or infinite set of Operands
$ ~* q$ O: J2 X" n4 b (called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator1 T# q- \- T! n
to form well-defined products and which furthermore satisfy the following conditions:
! p: e8 b) {( V5 V6 o1. Closure: If and are two elements in , then the product is also in .
& \3 A5 l% c! A% [. \8 E2. Associativity: The defined multiplication is associative, i.e., for all , . ( c. n! [. D" U0 v! d+ O
3. Identity: There is an Identity Element
' _+ U7 J6 C* R$ g: g (a.k.a. , , or ) such that for every element .
" Z5 }0 E/ k4 y$ M9 E4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of . & Q- M) d! Z+ L  p, e
A group is therefore a Monoid  q& Q: e! M9 _4 l% K
for which every element is invertible. A group must contain at least one element.
5 |* x% s+ u* b1 t# @( d * n. z3 `3 T4 ~. o! G
The study of groups is known as Group Theory
) Z; Y1 d" g; G% N. If there are a finite number of elements, the group is called a Finite Group
# o# B9 Z" z. x: ?* ] and the number of elements is called the Order
% ^& D* d; N9 Q. H/ u" K. q of the group. 7 @! A0 Q* r& y3 ~4 H) K' K! h9 B
( {$ h: C) w6 ]3 F4 g5 j
Since each element , , , ..., , and is a member of the group, group property 1 requires that the product
# d1 v( q& b& J/ Z; y' ]1 N
; O/ [  @7 F, l4 C: b. J; p& E; U9 u
(1)
% c& J8 g; L1 e- C+ T
) {" Y& ~( g- I0 U
# U! _3 M; n) D  {  h. }8 m* @

( V. l# N8 D# B1 Y8 |* Y+ l! @must also be a member. Now apply to ,
8 g% U& u9 z  o2 @% {9 _$ D . ~0 m& }4 i8 J
- [3 G! V1 @: L6 {3 v
(2)
9 u; a0 D" y7 Z& r/ N4 V

9 m) s( Z, d% V9 @- E# O. e2 \' y/ H6 C; d- w$ _

" B( v: P  J0 Q7 C1 W3 p' Y, kBut
5 U3 G5 @  R9 C$ g
# q* x+ F+ b1 g( y
) g0 Q9 L4 O& e
4 H. x& W) Y0 d& J+ [& j
0 D: H* r( ?9 K7 E' {8 o( Y0 f: h

- Y8 R! p; z* p; f# ~# b		(3)2 ?6 ?0 B% p9 D1 }2 w
so
" I) \) y  _" C0 ]
* g2 N) x. d5 L, a, y5 k
(4)! M  c9 D' ^% c  f, E% ]9 j

7 e. ^% F3 ~* }. o5 m- E8 i- a& M+ M3 X; d5 d9 |
" s" U- ]! x! Y) \- o) s' d
which means that + p6 `5 s! \. c/ n8 T( C7 \

1 `5 ~. X2 p7 f: r5 k1 Z		(5); [4 X4 `& H4 l0 h  v

+ u% p- B# S, z3 z, {8 Y7 W' w) E* Y7 v
* t/ L+ b1 f4 h( z( Q
1 E0 l, n% U7 n  J( ?0 e: G- nand
) N- G- Z3 F5 [6 q. s

( s5 L% f' x7 s/ K* ?$ J		(6)
5 S- q7 E/ j# l" w

0 I9 E9 m5 M5 _  |7 Q( i- ^2 x2 j0 N( n" M! E/ s; C% P+ [

, W  P" ]* s$ d$ g0 ~ , j1 R  F% s! |( B9 Y0 B

( _! F# |9 G# \5 ]! F, q
作者: OLS    时间: 2009-2-2 22:29
这个帖子乱。



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