Orders of products of elements and nilpotency of terms in the lower central series and the derived series

– In this paper we prove that if 𝐺 is a finite group, then the 𝑘 -th term of the lower central series is nilpotent if and only if for every 𝛾 𝑘 -values 𝑥, 𝑦 ∈ 𝐺 with coprime orders, either 𝜋 ( 𝑜 ( 𝑥 ) 𝑜 ( 𝑦 )) ⊆ 𝜋 ( 𝑜 ( 𝑥𝑦 )) or 𝑜 ( 𝑥 ) 𝑜 ( 𝑦 ) ≤ 𝑜 ( 𝑥𝑦 ) . We obtain an analogous version for the derived series of finite solvable groups, but replacing 𝛾 𝑘 -values by 𝛿 𝑘 -values. We will also discuss the existence of normal Sylow subgroups in the derived subgroup in terms of the order of the product of certain elements.