Multiplication Table

Before showing you a table, we ask, “what do you mean by multiplication?”

If you answer “3 and 4 give you 12”, then we know your perspective, and we give you a multiplication table example below:

\begin{bmatrix} * && 1 && 2 && 3 && 4 \\ 1 && 1 && 2 && 3 && 4 \\ 2 && 2 && 4 && 6 && 8 \\ 3 && 3 && 6 && 9 && 12 \\ 4 && 4 && 8 && 12 && 16 \end{bmatrix}

We could close the discussion, but now might be a good place to mention that the word “multiplication” gets used for more than just “12 from 3 and 4”. When symmetry operations are done on an object, more specifically, when we do one operation and then we follow it by another operation (examples: rotation, reflection, etc.) we say that we are multiplying the operations. An example is shown below:

\begin{bmatrix} && E && C_3^1 && C_3^2 \\ E && E && C_3^1 && C_3^2 \\ C_3^1 && C_3^1 && C_3^2 && E \\ C_3^2 && C_3^2 && E && C_3^1 \end{bmatrix}

We could agree calling the first example we showed you a Multiplication Table and calling the second example a Cayley Table, but if you go read about a Cayley Table, you will find that a Multiplication Table is an example of a Cayley Table, so we see what we call a “criss-cross” of definitions.

Our goal at this point is for you to be aware of both names, Multiplication and Cayley, and know which one is a subset of the other. And yes, even to not panic, should in your reading you come across an author who doesn’t follow what we said above.