论文标题
在言语大的小组中
On groups with large verbal quotients
论文作者
论文摘要
令$ w = w（x_1，...，...，x_n）$是一个单词，即免费组的元素$ f = \ langle x_1，...，x_n \ rangle $。组$ g $的语言亚组$ w（g）$是由$ \ {w（x_1，...，...，x_n）生成的子组：x_1，...，...，x_n \ in G \} $ in g $ in $ g $中的所有$ w $ values。按照J.González-Sánchez和B. Klopsch的遵循，如果$ | h：h：w（h）| <| g：w（g）| $每$ h <g $。在本文中，我们对$ w $ - 毫米群给出了新的结果，并研究了以前不平等并不严格的较弱状况。给出了一些应用程序：例如，如果有限的组具有尺寸$ n $的可解决（分别为nilpotent）部分，则它具有一个至少$ n $的可解决（分别为nilpotent）大小的可解决（nilpotent）子组。
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 $n$.
