标题: Group [打印本页] 作者: OLS 时间: 2009-2-2 22:25 标题: Group Group 4 X: X; ^, o# H" X, A' i# ], rA group is defined as a finite or infinite set of Operands ) _" J# [) X: u9 p. z; t4 P9 k& Y (called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator 3 K8 W; X" `9 j5 B, z) x* h to form well-defined products and which furthermore satisfy the following conditions: & }$ d( A, y' H _
1. Closure: If and are two elements in , then the product is also in . $ [% o* U8 q0 i8 b! f; k2. Associativity: The defined multiplication is associative, i.e., for all , . % E d# i4 z0 C4 B
3. Identity: There is an Identity Element* p2 P0 y3 s6 r* E
(a.k.a. , , or ) such that for every element . 3 S: h& H6 c8 e( K9 b8 A0 {2 J4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of . # \$ [; s' o `/ l6 x+ W) ^A group is therefore a Monoid1 w( f! u# j$ u2 \
for which every element is invertible. A group must contain at least one element. 7 q% T% L) Y3 [" G. T& A
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The study of groups is known as Group Theory 6 {4 a7 }8 @) @1 Y$ j2 f; `. If there are a finite number of elements, the group is called a Finite Group# k/ B8 R9 c2 \& [% |3 J
and the number of elements is called the Order 7 S% @% p5 ?% z of the group. 1 `0 u* c+ c' h8 c. E$ }
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Since each element , , , ..., , and is a member of the group, group property 1 requires that the product 9 {- @" V+ Y6 Q f) D$ p1 Z