# Group Theory

A **group** is a set of elements $$Z = {a, b, c, ...}$$ and one binary operator $$+$$ that satisfies the following axioms:

* **Closure:** for any$$a,b ∈ Z$$, the element $$a + b$$ is in $$Z$$ .
* **Associativity:** for any $$a,b,c ∈ Z$$, $$(a + b) + c = a + (b + c)$$.
* **Identity:** $$a + I = a$$ , for all $$a ∈ Z$$.
* **Invertibility:** $$a + (-a) = I$$, for all $$a ∈ Z$$.

In addition to the above properties, if a group exhibits the commutative property of $$a + b = b + a$$, it is called an **abelian group**.

{% hint style="info" %}
The elliptic curve used in Bitcoin is actually a mathematical group, that is **finite**, **cyclic**, **abelian**, and has a **single-generator** point, defined over the **binary addition operator**. These properties form the bedrock for an efficient signature and verification mechanism in Bitcoin.
{% endhint %}

A single-generator group contains an element $$G ∈ Z$$, called the **generator point**, such that repeated additions of $$G$$ with itself can generate every element in $$Z$$.

{% hint style="info" %}
In fact, in prime order elliptic curves, any point can be a generator point.
{% endhint %}

$$Z = {G, 2G, 3G, 4G, ...}$$ 

Additionally, our group is cyclic, which means it has an order $$n$$, such that $$nG = I$$.

Let us now represent the generator point $$G$$in Python, used in Bitcoin.

```python
# Generator point of the abelian group used in Bitcoin
G = Point(
    x=0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798,
    y=0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8,
    curve=secp256k1
)

# Order of the group generated by G, such that nG = I
N = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
```

#### Resources

* <https://web.stanford.edu/class/ee392d/Chap7.pdf>