Summary: Free knots, which were introduced by V. G. Turaev under the name of homotopy classes of Gaussian words, are a very interesting object of knot theory. They are closely related to classical knots in nature, and studying them uses wide and natural points of view, such as Kauffman’s virtual knots, Goldman and Turaev’s Lie algebras and coalgebras of curves on 2-surfaces, and Il’yutko and V. O. Manturov’s graph-links. Their relation to the theory of classical knots is based on the general notions of chord diagrams, the three Reidemeister moves, and embedding of tetravalent graphs in 2-surfaces. Turaev conjectured that the free knots are trivial. This conjecture was first disproved by V. O. Manturov, who used the new notion of parity [Sb. Math. 201, No. 5, 693–733 (2010; Zbl 1210.57010)]. In this paper, we use the same notion of parity to construct a simple strong invariant of free knots taking values in some group.