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 $\mathbb{Z}^{*}_n$.
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。