  Let K be a an alternating prime knot in the 3-sphere. We investigate the
category of flypes between reduced alternating diagrams for K. As a
consequence, we show that any odd prime order action on K is isotopic through
maps of pairs to a single flype. This implies that for any odd prime order
action on K there is either a reduced alternating periodic diagram or a reduced
alternating free periodic diagram. Finally, we deduce that the quotient of an
odd periodic alternating knot is also alternating.
