论文标题
有限nilpotent群体的单词问题
Word problems for finite nilpotent groups
论文作者
论文摘要
让$ w $是$ k $变量中的一个单词。对于有限的nilpotent $ g $，阿米特的一种猜想指出，$ n_w（1）\ ge | g |^{k-1} $，其中$ n_w（1）$是$ k $ -tuples $（g_1，...，...，...，g_k）\ in G^{（k）$ w（g_1，g_k）目前，该猜想对于Nilpotency 2类都是正确的。在这里我们考虑了Amit的猜想的广义版本，并证明$ n_w（g）\ ge | g | g | g |^{k-2} $，其中$ g $是$ w $ v $ v $ g $ g $ in $ g $，对于$ g $ g $ g $ g $ g $ a niLperments a nilpers二级$ a。 $ n_w（g）\ ge | g | $，其中$ g $是$ w $ - $ g $ in $ g $的有限群$ g $ nilpotency 2类。此外，对于$ p $ a Prime，我们表明有限的$ p $ - groups $ g $，带有两个独特的不可舒服的复杂性特征$ $ $ $ $ $ $ wime $ wime $ w _ = [x_1，y_1] ... [x_k，y_k] $ with $ k $ a自然号码;也就是说，对于$ g $ a $ w_k $ -value in $ g $，我们有$ n_ {w_k}（g）\ ge | g |^{2k-1} $。 最后，我们讨论了合理和手性的相关群体属性，并表明Nilpotency 2类的每个有限群都是合理的。
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.
