I am having trouble with the following passage of "Algebra and Geometry" by Alan F. Beardon in the highlighted part. I've read it several times but I do not understand what he is saying. Can you give me more explanation? Perhaps a couple of examples? Thanks.
I am thinking of the highlighted part as:
If x is an element of orbit j, then the cycle permutation j on x is equal to the original permutation of x. Meaning that cycles over elements that are parts of orbits will give the same outcome as original permutations would have given over the same element.
From the answer of Shubham I think that the second part says that a cycle will only fix an element if it is disjoint from other cycles. Is "i ≠ j" meaning disjoint? I thought that disjoint and not equal were not the same notions.