here is another interesting question that I can't answer on my own.

Let G be a countable, discrete group such that for any subgroup H of G and any element s of G we have: [H : sHt] is finite and [H : tHs] is finite where st=ts=1 i.e. t is the inverse of s. Then G is amenable.

Note that in the case when the indexes above are exactly equal 1 then G is Hamiltonian which is solvable and therefore amenable.

Thanks in advance.