📜  群的基本性质

📅  最后修改于: 2021-09-22 10:57:27             🧑  作者: Mango

令集合 G 从一个群 (G , o) 定义二元运算 o 。如果 G 满足以下 3 个性质,则 G 是一个群:

  • 关联性
  • 身份

组的属性:

属性-1
如果 a , b, c ∈ G 那么,是 aob = aoc ⇒ b = c

证明: –

Given a o b = a o c, for every a, b, c ∈  G  
Operating on the left with a-1, where a-1 ∈ G we have 
      a-1 o (a o b) = a-1 o (a o c) 
or  (a-1 o a) o b = (a-1 o a) o c         [using associative property]
or   e o b = e o c,                       [using inverse property]
or      b = c,                            [using identity property]

注意 aob 也写成 ab。

这被称为左抵消定律。

属性-2:
对于每个 a ∈ G ,eoa = a = aoe,其中 e 是单位元素。即左标识元素也是右标识元素。

证明: –

If a-1 be the left inverse of a, then 
              a-1 o (a o e) = (a-1 o a) o e           [using associative property]
 or         a-1 o (a o e) = e o e                     [using inverse property]
                                 = e                  [using identity property]
 or         a-1 o (a o e) = a-1 o a                   [using inverse property]
 i.e.        a-1 o (a o e) = a-1 o a  

因此, aoe = a by property-1即左消消法。因此我们发现 e 也是正确的单位元,所以它只被称为单位元。

属性 3:
对于每个 a ∈ G , a -1 oa = e = aoa -1即一个元素的左逆也是它的右逆。

证明: –

a-1 o (a o a-1) = (a-1 o a) o a-1    [using identity property]
      = e o a-1                                  [using inverse property]
      = a-1 o e                              [by property 2]
 i.e. a-1 o (a o a-1)= a-1 o e
Hence, a o a-1 = e, by left cancellation law. 

因此,我们发现元素 a 的左逆 a -1也是它的右逆,因此 a -1仅称为 a 的逆。

属性 4:
如果 a , b, c ∈ G 那么,是 boa = coa ⇒ b = c      

证明: –

Given a o b = a o c, for every a, b, c ∈  G  
Operating on the left with a-1, where a-1 ∈ G we have
       (b o a) o a-1 =  (c o a) o a-1
or      b o (a-1 o a)  = c o (a-1 o a)           [using associative property]
or      b o e = c o e,                           [using inverse property]
or      b = c,                                   [using identity property]

这被称为权利取消法。

属性-5:
对于每个 a , b ∈ G 我们有 (aob) -1 = b -1 oa -1即 G 组的两个元素 a、b 的乘积(或复合)的逆是取相反顺序的两个元素的倒数。

证明: –

Let a-1 and b-1 be the inverses of a and b. 
Now,(a o b) o (b-1 o a-1) = a o (b o b-1)  o a-1        [using  associative property]
= a o e o a-1                                  [using inverse property]
= a o a-1                                       [using identity property]
= e                                                [using inverse property]
(a o b) o (b-1 o a-1) = e
Similarly, (b-1 o a-1) o ( a o b)= e

因此,由逆 b -1 oa -1的定义是 ao bie (aob) -1 =b -1 oa -1 的逆

这被称为反转规则。