arXiv 2005.036342020-05-15math.GR

有限幂零群的字问题

Word problems for finite nilpotent groups

Rachel D. Camina, Ainhoa Iniguez, Anitha Thillaisundaram

备注：: 9 pages

摘要：

让$w$是$k$变量中的一个单词。对于有限幂零群$G$，Amit的一个猜想指出，$N_w(1) \ge |G|^{k-1}$，其中$N_w(1)$是$k$ -元组的个数$(g_1,...,g_k)\in G^{(k)}$使得$w(g_1,...,g_k)=1$ .目前，这个猜想已知对于幂零类2的群是正确的。本文考虑Amit猜想的一个推广，证明了对于幂零类2的奇数阶有限群$G$，有$N_w(g) \ge |G|^{k-2}$，其中$g$是$G$中的$w$ -值.如果$w$是一个字在两个变量，我们进一步表明，$N_w(g) \ge |G|$，其中$g$是$w$ -值在$G$有限群$G$的幂零类2。此外，对于$p$一个素数，我们表明，有限$p$ -组$G$，有两个不同的不可约复杂的字符度，满足广义阿米特猜想的话$w_k =[x_1,y_1]...[x_k,y_k]$与$k$一个自然数，也就是说，对于$g$一个$w_k$ -值在$G$，我们有$N_{w_k}(g) \ge |G|^{2k-1}$。最后，我们讨论了群的有理性和手征性的相关性质，证明了幂零类2的有限群都是有理群。

英文摘要：

Let $w$ be a word in $k$ variables. For a finite nilpotent group $G$, a conjecture of Amit states that $N_w(1) \ge |G|^{k-1}$, where $N_w(1)$ is the number of $k$-tuples $(g_1,...,g_k)\in G^{(k)}$ such that $w(g_1,...,g_k)=1$. Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit's conjecture, and prove that $N_w(g) \ge |G|^{k-2}$, where $g$ is a $w$-value in $G$, for finite groups $G$ of odd order and nilpotency class 2. If $w$ is a word in two variables, we further show that $N_w(g) \ge |G|$, where $g$ is a $w$-value in $G$ for finite groups $G$ of nilpotency class 2. In addition, for $p$ a prime, we show that finite $p$-groups $G$, with two distinct irreducible complex character degrees, satisfy the generalized Amit conjecture for words $w_k =[x_1,y_1]...[x_k,y_k]$ with $k$ a natural number; that is, for $g$ a $w_k$-value in $G$ we have $N_{w_k}(g) \ge |G|^{2k-1}$. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文