A short proof of the Khukhro--Makarenko theorem on large characteristic   subgroups with laws
  We give a short proof and some strengthening of the Khukhro--Makarenko theorem that each group virtually satisfying an outer commutator identity contains a finite-index characteristic subgroup satisfying this identity. An estimate for the index of this characteristic subgroup is obtained. 
Anton A. Klyachko, Yulia B. Melnikova