Multiplication Table

Notes

Group Theory

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

\begin{array}{ccc} e & g_{1} & g_{2} & \dots & g_{n} \\ e & e & g_{1} & g_{2} & \dots & g_{n} \\ g_{1} & g_{1} & g_{1}^{2} & g_{1} g_{2} & \dots & g_{1} g_{n} \\ g_{2} & g_{2} & g_{2} g_{1} & g_{2}^{2} & \dots & g_{2} g_{n} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ g_{m} & g_{m} & g_{m} g_{1} & g_{m} g_{2} & \dots & 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.

Group Order Group Mappings