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 $.