Large and symmetric: The Khukhro–Makarenko theorem on laws — without laws

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. Also, we answer a question of Makarenko and Shumyatsky.