3 Groups and Construction of Integers
3.1 Groups
We now define the concept of a group. A group is an algebraic structure, which helps us to better understand the abstract properties of the different sets of numbers defined above.
We start with what is called a binary operation, i.e. an operation such as \(+,\cdot,/,-\)
Definition 3.1 • Binary Operation
Let \(M\not = 0\) be a set. We call a mapping \[\tag{3.1} \begin{aligned} \circ\colon M\times M\longrightarrow M\\ \nonumber (a,b)\ni M\times M \mapsto \circ (a, b)\in M \end{aligned}\] a binary operation on \(M.\) We usually write \(a\circ b\) instead of \(\circ (a, b).\)
If for the binary relation \(\circ\) it holds \[\tag{3.2} \circ (a,b) = \circ (b,a)\, ,\] we call \(\circ\) commutative.
Examples of binary operations are the usual addition “\(+\)” or the multiplication “\(\cdot\)”. On \(M=\mathbb{N},\) \(\circ=+\) is a binary operation and we write, e.g. \(\circ(2,3)=+(2,3)=2+3.\)
Since the co-domain of \(\circ\) is \(M,\) we call the binary operation closed.
Here you can find the video explanation on \(\mathbb{N}, \mathbb{Z}\) and groups.
Definition 3.2 • Group
A group is a set \(G\not = \emptyset\) equipped with a binary operation \(\circ\colon G\times G\longrightarrow G\) such that we have
For all \(a,b,c\in G\) it holds Associativity \[\tag{3.3} a\circ (b\circ c) = (a\circ b)\circ c\, .\]
There exists \(e\in G\) such that Neutral Element \[\tag{3.4} e\circ a = a\circ e = a\quad\forall a\in G\, .\]
For every \(a\in G\) there exists \(b\in G\) such that Inverse Element \[\tag{3.5} a\circ b = b\circ a = e \, .\] We call \(b\) the inverse element to \(a\) and write \(b=a^{-1}.\)
3.2 Construction of Integers
Here you can find the video on construction of \(\mathbb{Z}.\)
Here you can find the video explanation on fields.
For completeness, we also include the definition of a field here.
Definition 3.3 • Field
Let \(\mathcal{F} \neq \emptyset\) be a set equipped with the binary operations \(+ \colon \mathcal{F} \times \mathcal{F} \longrightarrow \mathcal{F}\) and \(\cdot \colon \mathcal{F} \times \mathcal{F} \longrightarrow \mathcal{F}\) with respective neutral elements 0 and 1. We call \(\mathcal{F}\) a field if
\((\mathcal{F}, +)\) is a commutative (or Abelian) group.
\((F \setminus \{0\}, \cdot)\) is a commutative (or Abelian) group.
The following distributive law holds: \[a \cdot (b + c) = a \cdot b + a \cdot c \qquad \forall a, b, c \in \mathcal{F}.\]