Group
A non-empty set \(G\) is called a group when \(G\) is endowed with a binary operation * subject to some possibly familiar conditions. This binary operation is a mapping:
\begin{equation*} \star : G \times G \rightarrow G \end{equation*}which assigns to every pair \((x,y)\), in the Cartesian product G × G, an element x ⋆ y in G called the composite of x with y. This x ⋆ y is also known as the product of x with y in G.
The binary operation of composition must follow three laws.
• The associative law:
\begin{equation*} (x \star y) \star z = x \star (y \star z), \forall x,y,z \in G \end{equation*}• The neutral element requirement:
\begin{equation*} \exists e \in G : e \star x = x \star e = x, \forall x \in G \end{equation*}Such an \(e\) is called a neutral element of \(G\).
• The inverse element requirement:
\begin{equation*} \forall x \in G, \exists y \in G : x \star y = y \star x = e \end{equation*}Such a \(y\) is called an inverse for \(x\).