# Thursday August 15th

> We'll be using Hungerford's Algebra text.

## Definitions

The following definitions will be useful to know by heart:

- The order of a group
- Cartesian product
- Relations
- Equivalence relation
- Partition
- Binary operation
- Group
- Isomorphism
- Abelian group
- Cyclic group
- Subgroup
- Greatest common divisor
- Least common multiple
- Permutation
- Transposition
- Orbit
- Cycle
- The symmetric group $S_{n}$
- The alternating group $A_{n}$
- Even and odd permutations
- Cosets
- Index
- The direct product of groups
- Homomorphism
- Image of a function
- Inverse image of a function
- Kernel
- Normal subgroup
- Factor group
- Simple group

Here is a rough outline of the course:

- Group Theory
  - Groups acting on sets
  - Sylow theorems and applications
  - Classification
  - Free and free abelian groups
  - Solvable and simple groups
  - Normal series
- Galois Theory
  - Field extensions
  - Splitting fields
  - Separability
  - Finite fields
  - Cyclotomic extensions
  - Galois groups
  - Solvability by radicals
- Module theory
  - Free modules
  - Homomorphisms
  - Projective and injective modules
  - Finitely generated modules over a PID
- Linear Algebra
  - Matrices and linear transformations
  - Rank and determinants
  - Canonical forms
  - Characteristic polynomials
  - Eigenvalues and eigenvectors

## Preliminaries

**Definition**:
A **group** is an ordered pair $(G, \wait: G\cross G \to G)$ where $G$ is a set and $\wait$ is a binary operation, which satisfies the following axioms:

1. **Associativity**: $(g_1 g_2)g_3 = g_1(g_2 g_3)$,

2. **Identity**: $\exists e\in G \suchthat  ge = eg = g$,

3. **Inverses**: $g\in G \implies \exists h\in G \suchthat gh = gh = e$.

*Examples of groups:*

- $(\ZZ, +)$

- $(\QQ, +)$

- $(\QQ\units, \times)$

- $(\RR\units, \times)$

- ($\GL(n, \RR), \times) = \theset{A \in \mathrm{Mat}_n \suchthat \det(A) \neq 0}$

- $(S_n, \circ)$

**Definition:**
A subset $S \subseteq G$ is a **subgroup** of $G$ iff

1. **Closure**: $s_1, s_2 \in S \implies s_1 s_2 \in S$

2. **Identity**: $e\in S$

3. **Inverses**: $s\in S \implies s\inv \in S$

We denote such a subgroup $S \leq G$.

*Examples of subgroups:*

- $(\ZZ, +) \leq (\QQ, +)$

- $\SL(n, \RR) \leq \GL(n, \RR)$, where $\SL(n, \RR) = \theset{A\in \GL(n, \RR) \suchthat \det(A) = 1}$

## Cyclic Groups

**Definition**:
A group $G$ is **cyclic** iff $G$ is generated by a single element.

*Exercise*:
Show
$$
\generators{g} = \theset{g^n \suchthat n\in\ZZ} \cong \intersect_{g\in G} \theset{H \mid H \leq G \text{ and } g\in H}
.$$

**Theorem:**
Let $G$ be a cyclic group, so $G = \generators{g}$.

- If $\abs{G} = \infty$, then $G \cong \ZZ$.

- If $\abs{G} = n < \infty$, then $G \cong \ZZ_n$.

**Definition**:
Let $H \leq G$, and define a **right coset of $G$** by $aH = \theset{ah \suchthat H \in H}$.

A similar definition can be made for **left cosets**.

**The "Fundamental Theorem of Cosets"**:
$$
aH = bH \iff b\inv a \in H \text{ and } Ha = Hb \iff ab\inv \in H
.$$

**Some facts:**

- Cosets partition $H$, i.e.
$$
b\not\in H \implies aH \intersect bH = \theset{e}
.$$

- $\abs{H} = \abs{aH} = \abs{Ha}$ for all $a\in G$.

**Theorem (Lagrange)**:
If $G$ is a finite group and $H \leq G$, then $\abs{H} \divides \abs{G}$.

**Definition**
A subgroup $N \leq G$ is **normal** iff $gN = Ng$ for all $g\in G$, or equivalently $gNg\inv \subseteq N$.
(I denote this $N \normal G$.)

When $N \normal G$, the set of left/right cosets of $N$ themselves have a group structure.
So we define
$$
G/N = \theset{gN \suchthat g\in G}
\text{ where }
(g_1 N)\cdot (g_2 N) \definedas (g_1 g_2) N
.$$

Given $H, K \leq G$, define
$$
HK = \theset{hk \mid h\in H, ~k\in K}
.$$

We have a general formula,
$$
\abs{HK} = \frac{\abs H \abs K}{\abs{H \intersect K}}.
$$

## Homomorphisms

**Definition**:
Let $G,G'$ be groups, then $\varphi: G \to G'$ is a **homomorphism** if $\varphi(ab) = \varphi(a) \varphi(b)$.

*Examples of homomorphisms*:

- $\exp: (\RR, +) \to (\RR^{> 0}, \wait)$ since
$$
\exp(a+b) \definedas e^{a+b} = e^a e^b \definedas \exp(a) \exp(b)
.$$

- $\det: (\GL(n, \RR), \times) \to (\RR\units, \times)$ since
$$\det(AB) = \det(A) \det(B).$$

- Let $N \normal G$ and define

\begin{align*}
\varphi: G &\to G/N \\
g &\mapsto gN
.\end{align*}

- Let $\varphi: \ZZ \to \ZZ_n$ where $\phi(g) = [g] = g \mod n$ where $\ZZ_n \cong \ZZ/n\ZZ$

**Definition**:
Let $\varphi: G \to G'$.
Then $\varphi$ is a **monomorphism** iff it is injective, an **epimorphism** iff it is surjective, and an **isomorphism** iff it is bijective.

## Direct Products

Let $G_1, G_2$ be groups, then define
$$
G_1 \cross G_2 = \theset{(g_1, g_2) \suchthat g_1 \in G, g_2 \in G_2} \text{ where } (g_1, g_2)(h_1, h_2) = (g_1 h_1, g_2 ,h_2).
$$

We have the formula $\abs{G_1 \cross G_2} = \abs{G_1} \abs{G_2}$.

## Finitely Generated Abelian Groups

**Definition**:
We say a group is **abelian** if $G$ is commutative, i.e. $g_1, g_2 \in G \implies g_1 g_2 = g_2 g_1$.

**Definition**:
A group is **finitely generated** if there exist $\theset{g_1, g_2, \cdots g_n} \subseteq G$ such that $G = \generators{g_1, g_2, \cdots g_n}$.

This generalizes the notion of a cyclic group, where we can simply intersect all of the subgroups that contain the $g_i$ to define it.

We know what cyclic groups look like -- they are all isomorphic to $\ZZ$ or $\ZZ_n$.
So now we'd like a structure theorem for abelian finitely generated groups.

**Theorem**:
Let $G$ be a finitely generated abelian group.

Then
$$G \cong \ZZ^r \times \displaystyle\prod_{i=1}^s \ZZ_{p_i^{\alpha _i}}$$
for some finite $r,s \in \NN$ where the $p_i$ are (not necessarily distinct) primes.

*Example*:
Let $G$ be a finite abelian group of order 4.

Then $G \cong \ZZ_4$ or $\ZZ_2^2$, which are not isomorphic because every element in $\ZZ_2^2$ has order 2 where $\ZZ_4$ contains an element of order 4.

## Fundamental Homomorphism Theorem

Let $\varphi: G \to G'$ be a group homomorphism and define
$$
\ker \varphi \definedas \theset{g\in G \suchthat \varphi(g) = e'}
.$$

### The First Homomorphism Theorem

**Theorem**:
There exists a map $\varphi': G/\ker \varphi \to G'$ such that the following diagram commutes:

```{=latex}
\begin{center}
\begin{tikzcd}
G \arrow[dd, "\eta"'] \arrow[rr, "\varphi", dotted] &  & G' \\
&  &    \\
G/\ker \varphi \arrow[rruu, "\varphi'"]             &  &
\end{tikzcd}
\end{center}
```

That is, $\varphi = \varphi' \circ \eta$, and $\varphi'$ is an isomorphism onto its image, so $G/\ker \varphi = \im \varphi$.

This map is given by
$$
\varphi'(g(\ker \varphi)) = \varphi(g)
.$$

*Exercise*: Check that $\varphi$ is well-defined.

### The Second Theorem

**Theorem**:
Let $K, N \leq G$ where $N \normal G$. Then
$$
\frac K {N \intersect K} \cong \frac {NK} N
$$

*Proof:*
Define a map

\begin{align*}
K &\mapsvia{\varphi} NK/N \\
k &\mapsto kN
.\end{align*}

You can show that $\varphi$ is onto, then look at $\ker \varphi$; note that
$$
kN = \varphi(k) = N \iff k \in N
,$$

and so $\ker \varphi = N \intersect K$.

$\qed$