Published by De Gruyter March 29, 2024

On groups with large verbal quotients

Francesca Lisi and Luca Sabatini

From the journal Journal of Group Theory

https://doi.org/10.1515/jgth-2023-0088

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Abstract

Suppose that w = w ⁢ ( x 1 , … , x n ) is a word, i.e. an element of the free group F = ⟨ x 1 , … , x n ⟩ . The verbal subgroup w ⁢ ( G ) of a group 𝐺 is the subgroup generated by the set { w ⁢ ( x 1 , … , x n ) : x 1 , … , x n ∈ 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 ) | for every 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.

Dedicated to our professor Carlo Casolo

Communicated by: Benjamin Klopsch

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Received: 2023-06-12

Revised: 2024-03-04

Published Online: 2024-03-29

Published in Print: 2024-11-01

On groups with large verbal quotients

Abstract

References

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