  This paper is devoted to prove the existence of $q$-periodic alternating
projections of prime alternating $q$-periodic knots. The main tool is the
Menasco-Thistlethwaite's Flyping theorem. Let $K$ be an oriented prime
alternating knot that is $q$-periodic with $q\geq 3$, i.e. $K$ admits a
symmetry that is a rotation of order $q$. Then $K$ has an alternating
$q$-periodic projection. As applications, we obtain the crossing number of a
$q$ -periodic alternating knot with $q\geq 3$ is a multiple of $q$ and we give
an elementary proof that the knot $12_{a634} $ is not 3-periodic; this proof
does not depend on computer computations as in "Periodic knots and Heegaard
Floer correction terms" by Stanilav Jabuka and Swatee Naik (arXiv:1307.5116
[math.GT]).
