# Groups in General

### Definition

The general theory on groups. A group is a tuple \( G = (M,\star) \) of a set \( M \) and a binary operator \( \star \) for which the following three axioms hold:

- Associativity: For every \( a,b,c \in G \) we have that \( a \star (b \star c) = (a \star b) \star c \).
- The existence of an identity element: We have one unique element \( 1 \in G \) such that for every \( a \in G \) we have
$$1 \star a = a \star 1 = a$$
- Inverse element: For every \( a \in G \) we have a unique \( a^{-1} \in G \) such that
$$
a \star a^{-1} = a^{-1} a = 1
$$

The second axiom ensures that no group is empty. If we furthermore for every \( a,b \in G \) have that
$$
a \star b = b \star a
$$
we say that $G$ is abelian or that $G$ is commutative.

### Examples

- The set \( \mathbb{Z} = \{0,\pm 1,\pm 2, ...\} \) with the operator + forms a group. Call it \( G = (\mathbb{Z},+) \). We have that the operator is associative. We have that \( a + 0 = 0 + a = a \) for all \( a \in G \). And we have some \( a^{-1} \in G \) for all \( a \in G \) such that \( a + a^{-1} = a^{-1} + a = 0 \). For example \( 3 + -3 = -3 + 3 = 0 \).