Burnside groups and n-moves for links

Let n be a positive integer. M. K. Dabkowski and J. H. Przytycki introduced the nth Burnside group of links which is preserved by n-moves, and proved that for any odd prime p there exist links which are not equivalent to trivial links up to p-moves by using their pth Burnside groups. This gives counterexamples for the Montesinos-Nakanishi 3-move conjecture. In general, it is hard to distinguish pth Burnside groups of a given link and a trivial link. We give a necessary condition for which pth Burnside groups are isomorphic to those of trivial links. The necessary condition gives us an efficient way to distinguish pth Burnside groups of a given link and a trivial link. As an application, we show that there exist links, each of which is not equivalent to a trivial link up to p-moves for any odd prime p.