Group

A group is a non-empty set $G$ together with a binary operation (group low) on $G$, here denoted $\cdot$, such that the following requirements, known as group axioms, are satisfied:

1. Closure

   $\forall a,b \in G: a \cdot b \in G$

2. Associativity

   $\forall a,b,c \in G: (a \cdot b) \cdot c = a \cdot (b \cdot c)$

3. Identity (neutral) element

   $\exist e \in G, \forall a \in G: a \cdot e = e \cdot a = a$. Such element $e$ is unique. It is called the identity or neutral element of the group.

4. Inverse element

   $\forall a \in G, \exist b: a \cdot b = e$, where $e$ is the identity element. For each $a$, the element $b$ is unique; it is called the inverse of $a$ and is commonly denoted $a^{-1}$.

In cases where the group operation is commutative, the group is called abelian.

A group $G$ is said to be cyclic if there is some group element $g$ such that all group elements can be generated by repeatedly multiplying $g$ with itself, i.e., if every element of $G$ can be written as $g^i$ for some positive integer $i$. Such an element of $g$ is called a generator for $G$. Any cyclic group is abelian.

The cardinality $|G|$ is called the order of $G$. A basic fact from group theory is that for any element $g \in G$, $g^{|G|} = 1_G$. This implies that when considering any group exponentiation, i.e., $g^l$ for some integer $l$, reducing the exponent $l$ modulo the group size $|G|$ does not change anything: for any integer $l$, if $z ≡ l \mod{|G|}$, then $g^l = g^z$.

A subgroup of a group $G$ is a subset $H$ of $G$ that itself forms a group under the same binary operation as $G$ itself. Another basic fact from group theory states that the order of any subgroup $H$ of $G$ divides the order of $G$ itself. A consequence is that any prime-order group $G$ is cyclic: in fact, each non-identity element $g \in G$ is a group generator. This is because the set $\{g, g^2, g^3, ... , \}$ of powers of $g$ is easily seen to be a subgroup of $G$, referred to as the subgroup generated by $g$. Since $g ≥ 1_G$, its order is an integer strictly between $1$ and $|G|$, and since $|G|$ is prime, the order must equal $|G|$. Hence, the subgroup generated by $g$ in fact equals the entire group $G$.

Examples of groups:

• The most basic example of a commutative group is the group with just one element $\{•\}$ and the group law $• · • = •$. We call it the trivial group.

• Residue classes $(Z_n, +)$ for arbitrary moduli $n$ is a commutative group with the neutral element $0$ and the additive inverse $n − r$ for any element $r ∈ Z_n$.