Word problems for finite nilpotent groups
| Autor: | Ainhoa Iniguez, Rachel Camina, Anitha Thillaisundaram |
|---|---|
| Přispěvatelé: | Thillaisundaram, Anitha [0000-0001-8768-2590], Apollo - University of Cambridge Repository, Thillaisundaram, A [0000-0001-8768-2590] |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
General Mathematics
Character theory Secondary 20D15 Group Theory (math.GR) 010103 numerical & computational mathematics Words 01 natural sciences Article Combinatorics FOS: Mathematics Primary 20F10 0101 mathematics G110 Pure Mathematics Mathematics Finite group Conjecture Group (mathematics) 4901 Applied Mathematics 010102 general mathematics 4904 Pure Mathematics Order (ring theory) 20F10 20D15 Nilpotent Amit’s conjecture 49 Mathematical Sciences Nilpotent group Mathematics - Group Theory Rational words Word (group theory) |
| Popis: | 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. 9 pages |
| Databáze: | OpenAIRE |
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