Group

$Group$ $x$

(G, \cdot)

is a group:

(G 1) \forall a, b, c \in G (a \cdot (b \cdot c) = (a \cdot b) \cdot c) (G 2) \exists e \in G, \forall a \in G (e \cdot a = a \cdot e = a) (G 3) \forall a \in G, \exists x \in G (a \cdot x = x \cdot a = e)

\cdot

is closed binary operation on

A

:

\cdot : A \times A \to A, (a, b) \mapsto \cdot (a, b)

a \cdot b := \cdot (a, b)

e is identity element of (G, \cdot) : \forall a \in G (e \cdot a = a \cdot e = a) x is inverse element of a in (G, \cdot) : a \cdot x = x \cdot a = e

Identity and inverse element are unique

(G, \cdot)

is an abelian group:

(G 4) \forall a, b \in G (a \cdot b = b \cdot a)

An order of

A

∣ A ∣

: the number of elements in

A

(G, \cdot)

is a finite group:

G

is a finite set

(G, \cdot)

is an infinite group:

G

is an infinite set