# 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.
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.
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 $Z$.
In fact, in prime order elliptic curves, any point can be a generator point.
$Z = \{G, 2G, 3G, 4G, ...\}$
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.
# Generator point of the abelian group used in Bitcoin
G = Point(
x=0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798,