On groups with large verbal quotients

Abstract Suppose that w = w ⁢ (x 1, …, x n) w=w (x₁, , x₍) is a word, i. e. an element of the free group F = ⟨ x 1, …, x n ⟩ F= x₁, , x₍. The verbal subgroup w ⁢ (G) w (G) of a group 𝐺 is the subgroup generated by the set w ⁢ (x 1, …, x n): x 1, …, x n ∈ G \w (x₁, , x₍): x₁, , x₍ G\ of all 𝑤-values in 𝐺. Following J. González-Sánchez and B. Klopsch, a group 𝐺 is 𝑤-maximal if | H: w (H) | | G: w (G) | H: w (H) H G H<G. In this paper, we give new results on 𝑤-maximal groups, and study the weaker condition in which the previous inequality is not strict. Some applications are given: for example, if a finite group has a solvable (resp. nilpotent) section of size 𝑛, then it has a solvable (resp. nilpotent) subgroup of size at least 𝑛.