问题1:什么是一个algebraic structures Table 3.1.Multiplication table for Zs 01234567 0 000 0 0 000 012 3 4 5 6 7 2 0 24 6 24 6 3 1 4 2 5 4 0 x 0 4 0 4 0 4 5 5 2 6 6 0 7 46 25 04 6 2 2 1 (Z8
问题1:什么是一个algebraic structures? (Z8,•)
运算及其性质 Proposition 3.1 Let Zn be the set of equivalence classes of the integers mod n and a,b,c∈Zn. 1.Addition and multiplication are commutative: 2.Addition and multiplication are associative: 3.There are both an additive and a multiplicative identity: 4.Multiplication distributes over addition: 5.For every integer a there is an additive inverse-a: 6.Let a be a nonzero integer.Then ged(a,n)=1 if and only if there erists a multiplicative inverse b for a (mod n);that is,a nonzero integer b such that ab≡1(modn
运算及其性质
第二例: Figure 3.2.Symmetries of a triangle B B identity id= A B A B Table 3.2.Symmetries of an equilateral triangle B 0 id P1 P21 23 rotation p1=(B B id id C P1 p2 1 2 13 AC C B P1 P1 P2 id 43 41 2 P2 P2 id P1 2 3 1 rotation 2 3 id P2 2 2 d 3 μ1 2 P1 a Q B B 等边三角形的对称变换(函数)在 函数复合运算上构成的代数系统 B refection B
第二例: 等边三角形的对称变换(函数)在 函数复合运算上构成的代数系统
Table 3.4.Multiplication table for U(8) 13 5 7 1 13 5 7 3 3 1 75 5 7 13 7 7 5 31 在U(8)系统中,我们看到了单位元、看到了逆元,看到了这个结构: 5 3
在U(8)系统中,我们看到了单位元、看到了逆元,看到了这个结构: 1 3 5 7
再看等边三角形变换系统 Table 3.2.Symmetries of an equilateral triangle id PI p2 41 243 id id P1 2 1 2 13 id U3 P1 2 3 1 2 p2 p2 P1 2 μ3 1 1 2 id a P2 2 g g 陶 2 Pr a
再看等边三角形变换系统 id ρ1 ρ2 μ1 μ2 μ3
群-一种“公理化”的代数系统 The law of composition is associative.Thet is 问题2:结合律 (aob)oc=ao(boc) 为什么会被放进 fora,b,c∈G. 群公理中? There exists an element e G,called the identity element,such that for any element a EG eoa=aoe=a. For each element a G,there exists an inverse element in G, denoted by a-1,such that aoa-1=a-loa=e. 注意:对于the integers mod n,加法一定构成群,乘法则未必
群 – 一种“公理化”的代数系统 注意:对于the integers mod n,加法一定构成群,乘法则未必。 问题2:结合律 为什么会被放进 群公理中?