Determining whether a group is cyclic from its Cayley Table

Question

How can we check if a group is cyclic by using its Cayley table? Further, how we find out the generators from the Cayley table?

Answer 1

For a cyclic group each row in the Cayley table is the row above shifted across once, with respect to some ordering of the elements. Let me illustrate this with two examples.

For $\mathbb{Z}_4$ use the natural order:

$$\begin{array}{c|lcr} \mathbb{Z}_4 & 0 & 1 & 2 & 3 \\ \hline 0 & 0 & 1 & 2 & 3 \\ 1 & 1 & 2 & 3 & 0 \\ 2 & 2 & 3 & 0 & 1 \\ 3 & 3 & 0 &1 & 2 \end{array} $$

However $\mathbb{Z}_7^×$ requires a less intuitive ordering:

$$\begin{array}{c|lcr} \mathbb{Z}_7^× & 1 & 5 & 4 & 6 & 2 & 3 \\ \hline 1 & 1 & 5 & 4 & 6 & 2 & 3 \\ 5 & 5 & 4 & 6 & 2 & 3 & 1 \\ 4 & 4 & 6 & 2 & 3 & 1 & 5 \\ 6 & 6 & 2 & 3 & 1 & 5 & 4 \\ 2 & 2 & 3 & 1 & 5 & 4 & 6 \\ 3 & 3 & 1 & 5 & 4 & 6 & 2 \\ \end{array} $$

Since all cyclic groups are isomorphic (and hence have the same Cayley table after reordering and relabeling of elements), asking whether a group is cyclic is equivalent to asking whether a Cayley table of the form above exists.

An element is a generator if it appears in the row below the identity in a Cayley table of the form above , e.g. 1 and 5 in the examples above.