### Contents/Index

@1.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 $3 + -3 = -3 + 3 = 0$.