Burnside structures of finite subgroups

We establish conditions guaranteeing that a group $B$ possesses the following property: there is a number $\ell$ such that if elements $w$, $x^{-1}wx$, $\dots$, $x^{-\ell+1}wx^{\ell-1}$ of $B$ generate a finite subgroup $G$ then $x$ lies in the normalizer of $G$. These conditions are of a quite special form. They hold for groups with relations of the form $x^n=1$ which appear as approximating groups for the free Burnside groups $B(m,n)$ of sufficiently large even exponent $n$. We extract an algebraic assertion which plays an important role in all known approaches to substantial results on the groups $B(m,n)$ of large even exponent, in particular, to proving their infiniteness. The main theorem asserts that when $n$ is divisible by 16, $B$ has the above property with $\ell=6$.