In 2002, Dabkowski and Przytycki defined the $n$-th Burnside group of a link, which is an invariant preserved by $n$-moves. It is generally difficult to determine if the $n$-th Burnside groups of a given link and of the corresponding trivial link are isomorphic. In this talk, we give a necessary condition for the existence of such an isomorphism. In 2007, Dabkowski and Sahi introduced a finer invariant preserved by $4$-moves, which is defined as a quotient of the link group of a link. We also give several necessary conditions for which the Dabkowski-Sahi invariant of a given link is isomorphic to that of the trivial link. This is a joint work with Haruko Miyazawa and Akira Yasuhara.