On groups with large verbal quotients

Suppose that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>w</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>w</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> w=w(x₁,…,xₙ) is a word, i.e. an element of the free group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>F</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">⟨</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy="false">⟩</m:mo> </m:mrow> </m:mrow> </m:math> F= x₁,…,xₙ . The verbal subgroup <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>w</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> w(G) of a group 𝐺 is the subgroup generated by the set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mi>w</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo rspace="0.278em" stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace="0.278em">:</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>x</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> \{w(x₁,…,xₙ):x₁,…,xₙ∈ G\} of all 𝑤-values in 𝐺. Following J. González-Sánchez and B. Klopsch, a group 𝐺 is 𝑤-maximal if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>H</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mi>w</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>H</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mo><</m:mo> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>G</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mi>w</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> </m:mrow> </m:math> H:w(H)< G:w(G) for every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo><</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> 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.