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 ' n. C* {9 M5 M% G& ~! L- Y; R# y
A group is defined as a finite or infinite set of Operands7 p0 S, [$ f. {8 T" a& W# H
(called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator6 l3 d% C0 J6 c: Z! v
to form well-defined products and which furthermore satisfy the following conditions:
3 d% X# }' C# j+ @  g1. Closure: If and are two elements in , then the product is also in .
# d: g  {" c; u+ U2. Associativity: The defined multiplication is associative, i.e., for all , .
8 @  x. ?/ j6 e2 s1 R: R3. Identity: There is an Identity Element/ b- ?0 u1 T2 J
(a.k.a. , , or ) such that for every element . ! A1 T& o$ c, a) J
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 . 7 z/ O4 c7 d, b. Q0 A( [  k; g
A group is therefore a Monoid/ [* a4 v( r6 j% Y
for which every element is invertible. A group must contain at least one element.
, M# S  D* B3 I ; c' [" y1 X1 f) _( x
The study of groups is known as Group Theory
3 ^$ e9 D: r& @, Q+ z* `$ T5 H/ C. If there are a finite number of elements, the group is called a Finite Group
2 I7 j2 C) q) S6 D and the number of elements is called the Order
' g4 V, z7 J4 k2 q7 p of the group. $ _: E+ V! n3 ?9 H) ^5 z  g% Q6 d
  t6 H1 i& v7 O  f; Z8 M- b* K  k
Since each element , , , ..., , and is a member of the group, group property 1 requires that the product 6 i+ t: h5 c# E4 Z
" P3 I, r; ?& D" D& M. q
(1)
! a1 F* s2 }& d* `! @
- \  ~  Q2 x% r' B# ?
' x% J# W" R9 M

9 S3 n3 |, {2 |must also be a member. Now apply to ,
  l  Q4 d$ ~+ ^1 s# J  R, p
  y5 V  A5 X  ?) C

- c9 p6 ?& |; ]( R		(2)
5 i' U; V% U9 A) X& _
- N, Q. x/ `& M9 a) z) C' _9 J2 A. q

6 `& r$ f. F( b' l5 j- b1 E% e2 B0 R9 H1 a# @" U9 e/ b
But
, ?7 L& }0 i) {/ @, V
9 ]7 V- {9 E! u& _8 X  P
( Y* [# `- k0 N- t3 P$ l! J% O
% O! I1 `9 u* d% Z' J% n
 2 e4 l+ R; ]/ E% H4 }

+ o( y' ~2 y4 B2 m8 S# l- ^4 [		(3)9 W$ o; X3 i2 z- t, T- v8 a
so 7 x5 r5 H: N! \+ H* B9 W

7 C, ^! E: z. d- a( a  ]		(4)
7 K$ ?3 w2 k5 v) S
; B: u% d( p$ _2 N9 u, @

% P+ \& c4 Q- e
1 o# z8 m2 |( [2 J  Hwhich means that 7 }6 z0 d3 H) A/ I  f
: K! [- j4 Y2 `% k: v, S0 t
(5)2 \5 E! f6 E: Y2 X; _1 e9 t
! m3 |! n4 C2 p5 B

9 d- w! m. _) }9 F0 F" a. |% t/ n4 M: W' A9 c3 k
and
# b* m6 S( k% P) _7 \" D* r

, }) G5 M8 h, E  ^! w( O		(6): b; J% v' _$ a  W
$ e  S9 p$ j$ U/ F

; B: C$ ]: ?$ ^9 g. F7 m" W5 k$ L4 p8 o. p6 {9 x0 P

+ E7 o3 Z+ d% u1 n' V" k
5 g( Z$ p  L8 H
作者: OLS    时间: 2009-2-2 22:29
这个帖子乱。



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