[jlev1]

If there were a single element that generated the whole group, the group would be abelian.

[Skeime]

But the question here is not looking for a generator, because it would be okay if some group elements are only reached during the application of the sequence. (For the sequence to be a generator, all group elements need to be reached at the end of some full application of the sequence.)

The Hamiltonian cycle sequence from the original post is not a generator, but it visits every state. The question is: Is there a significantly shorter sequence that (when repeated) does the same?

[madcaptenor]

We can give a concrete example that non-Abelian groups can satisfy this with S_3, which is the smallest non-Abelian group. Swap the first two elements; then swap the last two. Repeat three times. You get the sequence

123, 213, 231, 321, 312, 132, 123

[bmenrigh]

I think we can go even further than that? If a single element generates the whole group, doesn't that mean the group is cyclic?