Let G be a finite group. A coprime commutator in is any element that can written as [x,y] for suitable x,y∈G such π(x)∩π(y)=∅. Here π(g) denotes the set of prime divisors order g∈G. An anti-coprime an [x,y], where π(x)=π(y). The main results paper are follows. If |xG|≤n whenever x commutator, then has nilpotent subgroup n-bounded index. every x∈G, H nilpotency class at most 4 [G:H] and |γ4(H)| ...

We prove a criteria for nilpotency of left braces in terms the $\star$-central series and also discuss Noetherian braces, obtaining some their elementary properties. show that if finitely generated brace $A$ is Smoktunowicz-nilpotent, then additive multiplicative groups are likewise generated.