Groups

A group has a mathematical definition. It is a set \(G\)5 with elements \(\require{physics} G=\qty{g_1, g_2, \dots, g_n}\), together with an operation \(\cdot\) (which is often called the group multiplication), meeting the following axioms:

  1. closedness \[\begin{equation} \forall g_1, g_2 \in G: g_1 \cdot g_2 \in G \end{equation}\]

  2. associativity \[\begin{equation} \forall g_1, g_2, g_3 \in G: g_1 \cdot (g_2 \cdot g_3) = (g_1 \cdot g_2) \cdot g_3 \end{equation}\]

  3. identity element \[\begin{equation} \exists! \thinspace e \in G: \forall g \in G: e \cdot g = g \cdot e = g \end{equation}\]

  4. inverses \[\begin{equation} \forall g \in G: \exists! \thinspace g^{-1} \in G: g \cdot g^{-1} = g^{-1} \cdot g = e \end{equation}\]

If the axiom

  1. commutativity \[\begin{equation} \forall g_1, g_2 \in G: g_1 \cdot g_2 = g_2 \cdot g_1 \end{equation}\] is also met, the group is called Abelian.

Groups play a central role in many scientific disciplines, but here are some examples of groups.

  1. We often use the same symbol to denote the set of group elements and the actual group. I don’t think there is anything wrong with this, as the distinction is most often clear from the context.↩︎