On groups with large verbal quotients

Suppose that w = w ⁢ ( x 1 , … , x n ) w=w(x_{1},\ldots,x_{n}) is a word, i.e. an element of the free group F = ⟨ x 1 , … , x n ⟩ F=\langle x_{1},\ldots,x_{n}\rangle . 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(x_{1},\ldots,x_{n}):x_{1},\ldots,x_{n}\in 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 ) | \lvert H:w(H)\rvert<\lvert G:w(G)\rvert for every 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.