Groups with finite periodic part (Q5903951)

It is proved the theorem: Let G be a group with involutions. Then the periodic part of G is finite iff there exist elements \(x,y\in G\) such that: (1) For almost all elements \(y^ g\), \(g\in G\), the subgroup \(gr(x,y^ g)\) is finite. (2) x,y are points (in the sense of the author) of G and one of these two elements has order 2, the other order \(\neq 2\).