Suppose the kth iterateof
has a fixed point
which is not a fixed point of any
for
. Then k is a power of 2.
By hypothesisand
, 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
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for
. In particular
makes up a single cycle of the permutation
, hence k is a power of 2 by Theorem B.