Theorem M

Suppose the kth iterate of has a fixed point which is not a fixed point of any for . Then k is a power of 2.

Proof M

By hypothesis and , for . All the 's are distinct for , since implies , since is one-one and onto. Consequently one can pick m large enough so that all the residue classes are distinct, for , where . Now the action of is exactly that of the permutation , hence

for . In particular makes up a single cycle of the permutation , hence k is a power of 2 by Theorem B.