A rough introduction to groups

I’m currently taking an abstract algebra course, taught by the amazing but quirky Michael Thaddeus. On the first day of class, he told us that this would be a course primarily about groups, but that they would be like “the protagonist of a novel who doesn’t appear until the fifth chapter.” Ie, that we had a lot of foundational material to cover (sets, functions, number theory) before we were well equipped to talk about groups.

Well, we got to that part by now. And since so many group theory concepts are used in the Gang Lab work I do, I wanted to really codify everything I learned about them.

Groups

To start, how do we define a group?

Def I: The group

A group is a set $G$ equipped with a binary operation (2 args, 1 val)

$M:G\times G \to G$, denoted $M(g,h)=:g\cdot h$

Satisfying the following properties:

(1) Associative

\[\begin{align} & \forall g,h,k \in G, \\ & g \cdot (h \cdot k) = (g \cdot h) \cdot k \end{align}\]

(2) Identity exists

\[\begin{align} &\exists e \in G \text{ st. } \forall g \in G, \\ &e \cdot g = g \cdot e = g \end{align}\]

(3) All inverses exist

\[\begin{align} & \forall g \in G, \exists h \in G \text{ st. } \\ & g \cdot h = h \cdot g = e \end{align}\]

Def II: The abelian group

Abelian groups on the other hand are commutative with respect to their operation!

Eg, $\forall g,h \in G$,

\[g \cdot h = h \cdot g\]

Non-abelian groups are lowkey rare

In fact, a lot of the most straightforward groups are abelian, like the integers with addition, and the rationals with multiplication.

• $(\mathbb{Z}, +)$
• $\mathbb{Q}^X := (\mathbb{Q}\setminus {0}, \cdot)$

We learned these were commutative back when we were first taught addition and multiplication in the first place.

In contrast, a non-abelian example would be like the matrices under matrix multiplication.

• $(M_{n\times n}, \cdot)$

Thm: Product groups

A product group is the cartesian product of two groups, which by theorem, is also a group.

If $G,H$ are both groups, then so is $G \times H$ equipped with the operation \((g_{1},h_{1}) \cdot (g_{2}, h_{2}) = (g_{1}\cdot g_{2}, h_{1} \cdot h_{2})\)

(Which smelled like the underpinnings of vector spaces)

Properties of groups

Now that we’ve defined the group, we can prove a very basic set of properties about them.

Prop I: Cancellation

We can ‘cancel’ group elements similar to normal cancellation in arithmetic.

\[\begin{align} & gk=hk \quad \text{ or } \quad kg=kh \\ \implies & g=h \quad \text{ ('canceling' the k)} \end{align}\]

Prop II: Uniqueness of identity

As a wise man (Michael Thaddeus) once said, “Just like a man cannot serve two masters, a group cannot have 2 identity elements.”

Ie, if $f \in G$ is also an identity element, then $f=e$.

A funny proof

Our professor said that the proof of the above property was “funny”.

\[f = ef = e\]

Since $f$ is an identity, anything times $f$ should be itself. Ditto for $e$, so multiply them together and we get $f=ef=e$.

Prop III: Uniqueness of inverses

Each element in a group has a unique inverse.

\[\begin{align} & \forall g \in G, \text{ if }h_{1} \text{ and } h_{2} \text{ are both inverses of }g, \\ & \implies h_{1}=h_{2} \end{align}\]

Another funny proof:

This was the second in his list of funny proofs for the day.

\[h_{1} = h_{1}e = h_{1}(gh_{2}) = (h_{1}g)h_{2}=eh_{2}=h_{2}\]

After which, he told us to “go home if you don’t get the joke. Go home and meditate.”