The functiondefined by
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is periodic with period
. The induced function
is a permutation, and its order is a power of 2.
(sketch). The theorem is established by induction on k, using the inductive hypotheses:(1)
is periodic with period
for
. In fact
![]()
for
.
(3)
and
are periodic with period
.
(4)
is a permutation whose order divides
. Also
![]()
I omit the details.