# Artin Algebra

# Chapter 2　Groups

## 2.1　Laws of Composition

Consider a set $T=\{a,b\}$, there are four maps:
1. **Identity map** $i:T\to T$ defined by $i(a)=a$ and $i(b)=b$ ;
2. **Transpose map** $\tau:T\to T$ defined by $\tau(a)=b$ and $\tau(b)=a$ ;
3. Constant function $\alpha:T\to T$ defined by $\alpha(a)=a$ and $\alpha(b)=a$ ;
4. Constant function $\beta:T\to T$ defined by $\beta(a)=b$ and $\beta(b)=b$ .
---
Some properties about **inverse**:
1. If an element $a$ has both a left inverse $l$ and a right inverse $r$, then $l=r$ is the inverse of $a$ .
2. If $a$ is invertible, then ==the inverse is unique==.
3. If $a,b$ are invertible with inverses $a^{-1},b^{-1}$, respectively, then ==$(ab)^{-1}=b^{-1}a^{-1}$==.
4. An element $a$ may have a left inverse or a right inverse, though it is not invertible.
:::spoiler Proof
1. $l=l(ar)=(la)r=r$.
2. Let $b,b'$ be inverses of $a$, then $b=b(ab')=(ba)b'=b'$.
:::
---
:::danger
The notation $\frac{b}{a}$ is not advisable, unless the law of composition is commutative. Because it is not clear whether the fraction stands for $a^{-1}b$ or for $ba^{-1}$.
:::

## 2.2　Groups and Subgroups
:::success
++**Definition**++
A **group** is a set $G$ together with a law of composition that has the following properties:
* $(ab)c=a(bc)$, $\forall a,b,c\in G$. (The composition is associative)
* $G$ contains an identity element $1$.
* Every element of $G$ has an inverse.
:::
:exclamation: An **abelian group** is a group where the law of composition is commutative.

---

:::success
++**Proposition**++ (**Cancellation Law**)
Let $a,b,c$ be elements of a group $G$ whose law of composition is written multiplicatively. If $ab=ac$ or if $ba=ca$, then $b=c$. In particular, if $ab=a$ or if $ba=a$, then $a=1$.
:::
:::spoiler Proof
Multiply both sides of $ab=ac$ on the left by $a^{-1}$ to obtain $b=c$.
:::

---

:::success
++**Definition**++
The $n\times n$ general linear group is defined by

$$
GL_n=\{ n\times n\ \textrm{invertible matrices}\}
$$

We can also write $GL_n(\mathbb{R})$ or $GL_n(\mathbb{C})$.
:::

## 2.3　Subgroups of the Additive Group of Integers

## 2.4　Cyclic Groups

## 2.5　Homomorphisms

## 2.6　Isomorphisms

## 2.7　Equivalence Relations and Partitions

## 2.8　Cosets

## 2.9　Modular Arithmetic

## 2.10　The Correspondence Theorem

## 2.11　Product Groups

## 2.12　Quotient Groups