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 sw

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