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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?

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  • $\begingroup$ What do you mean by the generators? Do you mean some minimal generating set? $\endgroup$ Mar 11, 2016 at 19:43
  • $\begingroup$ I just want to ask can we check whether the group is cyclic by inspecting the Cayley table of the group? $\endgroup$ Mar 11, 2016 at 20:47
  • $\begingroup$ Well yes, as the table gives us all information about the multiplication. Whether we can do it in an easy way is another matter. $\endgroup$ Mar 11, 2016 at 20:49
  • $\begingroup$ So actually in the very special case where the number of elements is a power of $2$, we can do this. Picking a candidate is easy, as you need to pick one not appearing on the diagonal. And any element not appearing on the diagonal will be a generator if the group is cyclic (and checking if it is a generator is also easy, as that is all about squaring, which means jumping around on the diagonal). $\endgroup$ Mar 11, 2016 at 21:01

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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.

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