  We prove a generalisation of the Khukhro--Makarenko theorem on large
characteristic subgroups with laws. This general fact implies new results on
groups, algebras, and even graphs and other structures. Concerning groups, we
obtain, e.g., a fact in a sense dual to the Khukhro--Makarenko theorem. A
graph-theoretic corollary is an analogue of this theorem in which planarity
plays the role of a multilinear identity. We answer also a question of
Makarenko and Shumyatsky.
