Circular Permutation, same arrangements?

Question

In the circular permutation, imagine objects A, B, C, D are arranged in a circle in the order ABCD, you rotate the circle and now you have, DABC, rotate again, then its CDAB, again then its BCDA. (Please try to imagine it in a circle)

"Why are all these permutations the same?" What exactly makes them same that we divide by n=4 to consider only one of those permutations. Can you explain me the "logic/concept" behind this? I know its to be divided by 4, but why? How do you see all of them as same?

Answer 1

They are considered the same since the circular sequence is exactly the same that is

• $A\to B\to C\to D\to A\to ...$

then the circular permutations for $n=4$ elements are given by $4!/4$ and more in general they are $(n-1)!$.

Answer 2

We treat them the same because that's how we've defined equality among functions (and permutations are special types of functions, and cycles are special types of permutations). To illustrate:

These four functions have the same (unordered) sets of input-output pairs, so we consider these functions to be equal.

We choose this definition because it's useful in many contexts. However, there are contexts in which this is an undesirable definition (and we benefit from using a definition in which these are considered different), in which case, we don't use this definition.