2.3 Groups

A group $G$ refers to a set of operations $t\in G$ that are changing some data or objects. Elementary examples refer to reflections in space, turning some sides of a Rubik's cube, or translations in space, as illustrated in Figure 2.1. The subsequent action of two group elements $t_1$ and $t_2$ of $G$ is another (typically more complicated) transformation $t_3 \in G$. Analogous to the concatenation of functions, we write $t_3 = t_2 \circ t_1$, and we say $t_3$ is $t_2$ after $t_1$. The set of transformations forms a group iff it obeys the following rules.

Definition 2.6 Group
A set $(G, \circ)$ is called a group with operation $\circ: G \times G \to G$ when the following rules apply

a) The set is closed: $\quad \forall g_1, g_2 \in G : g_1 \circ g_2 \in G$.
b) The set has a neutral element: $\quad \exists e \in G \; \forall g \in G: e \circ g = g$.
c) Each element has an inverse element: $\quad \forall g \in G \; \exists i \in G : i \circ g = e$.
d) The operation $\circ$ is associative: $\quad \forall g_1, g_2, g_3 \in G : ( g_1 \circ g_2 ) \circ g_3 = g_1 \circ ( g_2 \circ g_3 )$.

Definition 2.7 Commutative Group
A group $(G, \circ)$ is called a commutative group when

e) the group operation is commutative: $\forall g_1, g_2 \in G : g_1 \circ g_2 = g_2 \circ g_1$.

Remark 2.5. Commutative groups are also denoted as Abelian groups.

When the group has a finite number of elements the result of the group operation can explicitly be specified by a group table. We demonstrate this by the smallest groups. The empty set can not be a group because it has no neutral element. Therefore the smallest groups have a single element and two elements. Both of these groups are commutative.

Example 2.8 Smallest groups

$(\{ n \}, \odot)$ comprises only the neutral element.

$\odot$	$n$
$n$	$n$

The smallest non-trivial group has two elements $(\{ 0, 1\}, \oplus)$ with

$\oplus$	$0$	$1$
$0$	$0$	$1$
$1$	$1$	$0$

It describes the turning of a piece of paper:
- Not turning, $0$, does not change anything (neutral element).
- Turning, $1$, shows the other side, and turning twice is equivalent to not turning at all ($1$ is its own inverse).