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 $ Q* K9 S, ~% a" ?7 D/ Y
A group is defined as a finite or infinite set of Operands6 N4 d* o& y! ]/ V; {
(called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
% A/ ~$ \" ^- i. _/ C to form well-defined products and which furthermore satisfy the following conditions: 4 M" b( W. L4 c5 |3 [
1. Closure: If and are two elements in , then the product is also in .
1 Z# L+ |+ n) a% I" t- A5 o, x0 n0 Z2. Associativity: The defined multiplication is associative, i.e., for all , . - s$ W# B' d# S" y
3. Identity: There is an Identity Element
+ o7 \" ~, E" m* q7 X# N9 U5 } (a.k.a. , , or ) such that for every element .
) l8 ?( N, @8 |; c+ S' H4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of . / O( w( X/ n% R+ v
A group is therefore a Monoid
1 Z3 v5 @$ l3 g1 D# `* h# U for which every element is invertible. A group must contain at least one element. 7 x8 e1 S; p8 l$ W* f. g/ G! j# E% c3 ]

5 B! r' g7 k+ G0 I" k* m0 iThe study of groups is known as Group Theory% Y2 z6 R& s) a( H  t  W& E
. If there are a finite number of elements, the group is called a Finite Group5 `2 O" q2 U" z; ]2 i0 k
and the number of elements is called the Order
! R6 T! \8 _! b5 J; b9 {* y of the group.
& I4 V' m) P! J/ I
3 b6 h, |5 x7 gSince each element , , , ..., , and is a member of the group, group property 1 requires that the product   v- A7 v; w% s! [% k7 x
8 w% t. M  ^) U3 j$ c
(1)
% g2 e  J! D9 P4 ^, b
. N3 d$ u/ M7 e; z1 h+ Q

9 X2 B1 u2 K8 ~- c) v
7 u, i. B2 }/ @, P' r/ kmust also be a member. Now apply to , 0 [5 [8 p( O3 e+ d
* z# r" `* w5 X7 u, o6 {
6 ]& m( e( v" Y( a2 {/ J
(2), E, E9 `( y1 S) z/ ]$ B6 c
. K  B' D5 L! b7 h9 n5 O, L
1 b. K) K" J  w* I9 C0 M

- {$ v; o& q; p0 @4 G, h+ VBut
7 k+ T5 P: ~# N6 [3 [# H
$ g, ]/ U+ y0 @, {
 % p. n1 j" t' V0 y

% g5 q9 T+ H: K" g: N
- Y) A7 j1 \* C! W3 A: J
; A* r( n8 @& p
(3). N1 S5 T1 A' v) ~& s' ^) v0 `
so
* J& J* Q0 k- }. A; k

/ O. b8 n3 Y) @2 W7 Q		(4)
& j4 B. |5 ^- f. b7 M

9 @5 p) G7 `" T/ w& `3 v' C, [' N$ U5 K* p9 n' [  R8 @
2 g  E' H1 i$ [1 G- A
which means that + B- B- ~; E3 E. E+ J

+ O1 A' w$ s8 x( ^7 f1 E		(5)
4 k! h8 u* D4 X0 R8 \

9 P) v4 G8 r" l/ L
6 t7 i% }/ u2 V$ P6 w
' t3 I  J0 b) F- F: |& jand & E& |. r+ g) O; N
0 c- R" ]! p5 c7 ^/ `) D2 x! a. f' j
(6)! c9 b% o- K% u
( A" V% p) C$ |5 y9 E. d, p% O0 w

. d& b# D& u8 J7 P+ X) p5 C& N
4 R# |( M3 Q1 {) l # d2 B' h/ t( a# Z6 j4 `/ E

: G: C8 ^+ ]" v$ k9 ~# T$ [
作者: OLS    时间: 2009-2-2 22:29
这个帖子乱。



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