群论基础 — Lalande.

Group Theory: Introduction.

Group theory is the study of groups. Groups are sets equipped with an operation that satisfies certain basic properties. They are closure, identity, inverse, and associativity.

Set $\mathbb{G}$ and a binary operation $\circ$ defined on $\mathbb{G}$ that:

Closure: For all $g_1,g_2\in \mathbb{G}$ , it holds that $g_1 \circ g_2\in \mathbb{G}$
Identity: There exists an identity element $e\in \mathbb{G}$ s.t. $e\circ g=g=g \circ e$ for all $g\in \mathbb{G}$
Inverse: For every element $g\in \mathbb{G}$ there exists an inverse element $h\in \mathbb{G}$ s.t. $g\circ h=e=h\circ g$
Associativity: For all $g_1,g_2,g_3\in\mathbb{G}$ , $g_1\circ (g_2\circ g_3)=(g_1\circ g_2)\circ g_3$

Order of a finite group $\mathbb{G}$ , denoted $|\mathbb{G}|$ , is the number of elements in $\mathbb{G}$ .

If group $\mathbb{G}$ is an Abelian group, it will satisfiy the additionally commutativity: For all $g_1,g_2\in\mathbb{G}$ , $g_1\circ g_2=g_2\circ g_1$ .

Subgroups

Definition. If $\mathbb{G}$ is a group, a set $\mathbb{H}\subseteq \mathbb{G}$ is a subgroup of $\mathbb{G}$ if $\mathbb{H}$ itself forms a group under the same operation. $\mathbb{H}$ is a strict subgroup of $\mathbb{G}$ if $\mathbb{H} \neq \mathbb{G}$ .

Note that every group $\mathbb{G}$ always has the trivial subgroups $\mathbb{G}$ and $\{e\}$ .

Modulo Multiplication Group

A modulo multiplication group $(\mathbb{Z}/n)^{*}$ is a finite group of residue classes prime to $n$ under multiplication mod $n$ . $(\mathbb{Z}/n)^{*}$ is Abelian of group order $\varphi(n)$ , where $\varphi(n)$ is the totient function.

The multiplicative group of integers modulo $n$ , $(\mathbb{Z}/n)^{*}$ , is very worth researching.

$(\mathbb{Z}/n)^{*}:=$ invertible elements in $\{1,2,...n-1\}$ $:=$ $\{b\in \{1,...,n-1\}|\gcd(b,n)=1 \}$ .

Therefore $|(\mathbb{Z}/n)^{*}|=\varphi(n)$ . If $n$ is prime (which is usually denoted $p$ ), $|(\mathbb{Z}/p)^{*}|=p-1$ .

$(\mathbb{Z}/n)^{*}$ is also denoted $\mathbb{Z}^{*}_n$ .

Eureka moment!

群论虽然是第一次接触学习，但它的定义与相关的一些定理意外的容易理解 (遵循直觉)。可能是由于 "群" 这一概念在日常生活中已经渗透得很深，以致于抽象成为数学模型后也有较强的既视感。

这里想要重点提一下整数模 $n$ 乘法群 $(\mathbb{Z}/n)^{*}$ ，这是一个很重要的群。对于它的研究起源于对模 $n$ 剩余类 (residue classes) 的研究。我们知道，对于模 $n$ 剩余类 $\{0,1,2,...,n-1\}$ ，若群运算 (group operation) 是加法 $+$ ，剩余类能够很自然的形成一个群 $(\mathbb{Z}/n)^{+}$ 。然而，若我们将群运算设为乘法 $\times$ ，剩余类并不能形成一个群。这是因为群性质要求每个群元素都具有逆元 (inverse)；而在模 $n$ 剩余类中，只有与 $n$ 互质的群元素才有逆元。因此，我们将与 $n$ 不互质的元素剔除，剩下的就是整数模 $n$ 乘法群 $(\mathbb{Z}/n)^{*}$ 了。