### 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.