Groups

Let G be a non empty set. A function \circ : G \times G \to G is called a link on G. The pair (G, \circ) is called groupoid.

A function \circ is a link, if it is closed.

(G, \circ) is commutative if \forall a,b \in G: a \circ b = b \circ a The addition group (\mathbb{N}_{0},+) is a commutative (Albesch') groupoid. The subtraction group (\mathbb{N}_{0},-) is not a groupoid, because it's not closed. (2-7 \notin \mathbb{N}_{0})

Semigroups, Monoids, Groups

If \forall a,b,c \in (G, \circ): a \circ (b \circ c) = (a \circ b) \circ c, then is \circ associative, and (G, \circ) a semigroup.

If the semigroup G contains an e such that a \in G: e\circ a = a \circ e = a, then is e the neutral element, and G a Monoid

If \forall a \in G, \exists a^{-1} \in G: a \circ a^{-1}=e with G being a monoid, then G is a Group.

Finite Group

A groupoid (G_{n}, \circ) is called finite, if:

1. the set G is finite
2. \circ is a commutative group operation

Cyclical Groups

Let (G, \circ) be a group with e being the neutral element. For x\in G and z\in \mathbb{Z} we define: x^{z} = \left\{ \begin{array}{ll} x \circ (x^{z-1}) & \text{if } z > 0 \\ e & \text{if } z = 0 \\ x^{-1} \circ (x^{z+1}) & \text{if } z < 0 \\ \end{array} \right.

G is a cyclical group, if \exists a \in G: G = \{a^{z}| z \in \mathbb{Z} \}. Such an a \in G is called Producer or Generator of G.

Order

Let (G, \circ) be a group with e being the neutral element. a\in G is the smallest positive number z for which the following holds:

a^{z} = \underbrace{a \circ a \circ \dots \circ}_{z \text{ times}} = e

\langle {a} \rangle is the set \{a^{z} | z \in \mathbb{Z}\} and is called Subgroup. |\langle {a} \rangle| is the Order of the element a.