Some Engel conditions on finite subsets of certain groups (Q2771472)

Denote by \(E(n)\) (\(E_k(n)\)) the class of groups all of whose subsets consisting of \(n+1\) elements possess a pair \(x,y\) satisfying an Engel condition (satisfying the \(k\)-th Engel condition). -- Results: Every finitely generated soluble group and every finite group satisfying \(E(n)\) with \(n<3\) is nilpotent. Every finite group satisfying \(E(n)\) with \(n<16\) is soluble. \(S_3\) satisfies \(E(3)\); \(A_5\) satisfies \(E(16)\). -- Every finitely generated residually finite group \(G\) satisfying \(E_k(n)\) is finite by nilpotent; there is a positive integer \(c\) depending on \(k\) only such that \(G/Z_c(G)\) is finite. (Theorems 1.1-1.3).