On groups with large verbal quotients
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914711958716416 |
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| author | Lisi, Francesca Sabatini, Luca |
| author_facet | Lisi, Francesca Sabatini, Luca |
| contents | Let $w=w(x_1,...,x_n)$ be a word, i.e. an element of the free group $F = \langle x_1,...,x_n \rangle$. The verbal subgroup $w(G)$ of a group $G$ is the subgroup generated by the set $\{ w(x_1,...,x_n) : x_1,...,x_n \in G \}$ of all $w$-values in $G$. Following J. González-Sánchez and B. Klopsch, a group $G$ is $w$-maximal if $|H:w(H)| < |G:w(G)|$ 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 $n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_12021 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On groups with large verbal quotients Lisi, Francesca Sabatini, Luca Group Theory Primary 20F99 Let $w=w(x_1,...,x_n)$ be a word, i.e. an element of the free group $F = \langle x_1,...,x_n \rangle$. The verbal subgroup $w(G)$ of a group $G$ is the subgroup generated by the set $\{ w(x_1,...,x_n) : x_1,...,x_n \in G \}$ of all $w$-values in $G$. Following J. González-Sánchez and B. Klopsch, a group $G$ is $w$-maximal if $|H:w(H)| < |G:w(G)|$ 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 $n$. |
| title | On groups with large verbal quotients |
| topic | Group Theory Primary 20F99 |
| url | https://arxiv.org/abs/2203.12021 |