Mengzelev's Blog

问求学习笔记-群论初步

Word count: 1,223 / Reading time: 6 min
2019/03/02 Share

由于接下来的书没有中文版了,因此笔记中英文都看心情

整数等价类

命题3.4:模运算下的整数等价类具有以下性质:

  1. 加法和乘法交换律
  2. 加法和乘法结合律
  3. 加法(0)与乘法(1)恒等式
  4. 乘法分配律
  5. 任意元素都存在加法逆元
  6. $a$为非零整数,$gcd(a,n)=1$当且仅当$a$存在乘法逆元,即存在非零整数$b$,使得$ab\equiv 1(mod n)$

(虽然我觉得这玩意儿记了应该没什么卵用)
【复习时的我回来打脸了

定义

  • 二元运算(binary operation)或合成律(law of composition):a function $G\times G\to G$ that assigns to each pair $(a,b)\in G\times G$ a unique element $a\circ b$, or $ab$ in $G$, called the composition of $a$ and $b$
  • 群(group): a set $G$ together with a law of composition $(a,b)\mapsto a\circ b$ that satisfies the following axioms:
    1. 合成律满足结合律(asscociative)
    2. 存在单位元(identity element)$e\in G$,满足$e\circ a = a\circ e = a$
    3. 对于每个$a\in G$,都存在逆元$a^{-1}$,使得$a\circ a^{-1}=a^{-1}\circ a=e$
  • 群 = 运算封闭+结合律+单位元+逆元
  • 阿贝尔群(Abelian)或交换群(commutative):满足$a\circ b=b\circ a$的群,反之为nonabelian或noncommutative
  • 凯莱表(Cayley table):用加法或乘法描述群的表格
  • 可逆元素群(group of units): 拥有逆元的元素组成的群
  • 一般线性群(general linear group)
  • 四元群(quaternion group)
  • 群是有限的(finite),或者说有有限序数(has finite order),当它具有有限个元素,否则是无限的(infinite)或有无限序数(infinite order)

群的基本性质

命题3.17: 群中的单位元是唯一的
命题3.18: 逆元是唯一的
命题3.19: $G$是群,$a,b\in G$,则$(ab)^{-1}=b^{-1}a^{-1}$
命题3.20: $(a^{-1})^{-1}=a$
命题3.21: $ax=b$和$xa=b$在$G$中有唯一解
命题3.22(左右消去律right and left cancellation law): $ba=ca$ implies $b=c$ and $ab=ac$ implies $b=c$

对群中的元素,可以定义乘方

定理3.23: 在群中,一般指数的运算律成立

子群

  • 子群(subgroup):仿照子空间的定义
    • 平凡子群(trivial subgroup):$H={e}$
    • 真子群(proper subgroup)
  • 子群必须继承群的二元运算

子群相关的定理

命题3.30: $G$的子集$H$是子群当且仅当

  1. $G$的单位元$e\in H$
  2. $H$对$G$的运算封闭:If $h_1,h_2\in H$, then $h_1h_2\in H$
  3. If $h\in H$, then $h^{-1}\in H$

命题3.31: $H$是$G$的子群当且仅当$H\neq\emptyset$ and whenever $g,h\in H$ then $gh^{-1}$ is in $H$

循环子群(Cyclic Subgroups)

定理4.3: Let $G$ be a group and $a$ be any element in $G$. Then the set $\left⟨a\right⟩={a^k:k\in\mathbb{Z}}$is a subgroup of $G$. Furthermore, $⟨a⟩$ is the smallest subgroup of $G$ that contains $a$.

循环子群(Cyclic Subgroup):$⟨a⟩$
循环群(Cyclic group): 包含了某个元素$a$,使得$G=⟨a⟩$.此时$⟨a⟩$是$G$的生成器(generator).
The order of $a$: 最小的整数$n$满足$a^n=e$,表示为$|a|=n$。如果不存在满足要求的$n$,则称$a$是无穷的(infinite),表示为$|a|=\infty$
e.g. $\mathbb{Z}$和$\mathbb{Z}_n$都是循环群,1和-1是$\mathbb{Z}$的生成器,1是$\mathbb{Z}_n$的生成器但不一定是唯一的。

定理4.9: 所有循环群都是可交换的。(Every cyclic group is abelian).

定理4.10: 循环群的子群都是循环子群。(Every subgroup of a cyclic group is cyclic.)

引理4.11: The subgroups of $\mathbb{Z}$ are exactly $n\mathbb{Z}$ for $n=0,1,2…$

命题4.12: Let $G$ be a cyclic group of order $n$ and suppose that $a$ is a generator
for $G$. Then $a^k = e$ if and only if $n$ divides $k$.($n$能整除$k$,$k$能被$n$整除)

定理4.13: Let $G$ be a cyclic group of order $n$ and suppose that $a\in G$ is a generator
of the group. If $b = a^k$, then the order of $b$ is $n/d$, where $d = gcd(k, n)$.

引理4.14: The generators of $\mathbb{Z}_n$ are the integers $r$ such that $1\le r< n$ and
$gcd(r,n) = 1$. $\mathbb{Z}_n$的生成器与$n$互质。

复数乘法群(Multiplicative Group of Complex Numbers)

一堆复数的基础知识…….

$r(\cos\theta+i\sin\theta)$ 会被简写为 $r~cis\theta$

命题4.20: $z=r~cis\theta$ and $w=s~cis\phi$. Then $zw=rs~cis(\theta+\phi)$

圆群(The circle group)

定义: $$\mathbb{T}={z\in\mathbb{C}: |z|=1}$$

命题4.24: 圆群是$\mathbb{C}^*$的子群

定理4.25: If $z^n=1$, then the nth roots of unity are $$z=cis(\frac{2k\pi}{n})$$
where $k-0,1,…n-1$. Furthermore, the nth roors of unity form a cyclic subgroup of $\mathbb{T}$ of order $n$.

A generator for the group of the nth roots of unity is called a primitive nth root of
unity
.

重复平方法(The Method of Repeated Squares)

其实就是快速幂

理论基础: If $b\equiv a^x (\mod n)$ and $c\equiv a^y (\mod n)$, then $bc\equiv a^{x+y}(\mod n)$
$(a^{2n})^2\equiv a^{2\cdot 2n}\equiv a^{2^{n+1}}(\mod n)$

CATALOG
  1. 1.
    1. 1.1. 整数等价类
    2. 1.2. 定义
    3. 1.3. 群的基本性质
    4. 1.4. 子群
      1. 1.4.1. 子群相关的定理
  2. 2. 循环子群(Cyclic Subgroups)
    1. 2.1. 复数乘法群(Multiplicative Group of Complex Numbers)
      1. 2.1.1. 圆群(The circle group)
    2. 2.2. 重复平方法(The Method of Repeated Squares)