# 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:

1. Associativity: For every $$a,b,c \in G$$ we have that $$a \star (b \star c) = (a \star b) \star c$$.
2. The existence of an identity element: We have one unique element $$1 \in G$$ such that for every $$a \in G$$ we have $$1 \star a = a \star 1 = a$$
3. 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 $$3 + -3 = -3 + 3 = 0$$.