Multiplication Table

A useful construct for analyzing the properties of groups is the multiplication table. For a group $G$, of order $\lvert G \rvert = n-1 = m-1$, the multiplication table is

$$ \begin{array}{c|cc} & e & g_{1} & g_{2} & \cdots & g_{n} \\ \hline e & e & g_{1} & g_{2} & \cdots & g_{n} \\ g_{1} & g_{1} & g_{1}^{2} & g_{1}g_{2} & \cdots & g_{1}g_{n} \\ g_{2} & g_{2} & g_{2}g_{1} & g_{2}^{2} & \cdots & g_{2}g_{n} \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ g_{m} & g_{m} & g_{m}g_{1} & g_{m}g_{2} & \cdots & g_{m}g_{n} \end{array} $$

Note that the right-hand side of the product is a column element and, the left-hand side, a row element.