Characterisations of finite groups by subsets of commutators

Abstract Given a finite group 𝐺 and an element 𝑥 of 𝐺, let K x ⁢ ( G ) K_{x}(G) denote the set of commutators [ x , g ] [x,g] as 𝑔 varies through 𝐺. We prove that various group theoretical properties of 𝐺, for instance, nilpotency of several subgroups of 𝐺 and the existence of a Sylow tower for 𝐺, can be detected by considering various subsets of 𝐺 of the form K x ⁢ ( H ) K_{x}(H) where H ⩽ G H\leqslant G and x ∈ H x\in H .