Nilpotency and the number of word maps of a finite group

TL;DR

This paper characterizes the nilpotency of finite groups by analyzing the asymptotic growth of the number of word-induced functions as the number of variables increases.

Contribution

It introduces a novel characterization of nilpotency in finite groups based on the growth rate of word maps.

Findings

Nilpotent groups exhibit a specific growth pattern in the number of word maps.

The growth rate of $ ext{Ω}_d(G)$ distinguishes nilpotent groups from non-nilpotent ones.

Asymptotic analysis provides a new tool for understanding group structure.

Abstract

For a finite group $G$ and a non-negative integer $d$, denote by $\Omega_d(G)$ the number of functions $G^d\rightarrow G$ that are induced by substitution into a word with variables among $X_1,\ldots,X_d$. In this note, we show that nilpotency of $G$ can be characterized through the asymptotic growth rate of $\Omega_d(G)$ as $d\to\infty$.

Equations

$$w_f(X_1,\dots ,X_d):=\prod _{r=1}^d\prod _{i_1=1}^d\prod _{i_2=i_1+1}^d\cdots \prod _{i_r=i_{r-1}+1}^d[X_{i_1},\dots ,X_{i_r}{]}^{f(\{i_1,\dots ,i_r\})}.$$

$$[g_{i_1},\dots ,g_{i_r}{]}^{f(\{i_1,\dots ,i_r\})-g(\{i_1,\dots ,i_r\})}\ne 1_G.$$

$$w_f=w_0\cdot [X_{i_1},\dots ,X_{i_r}{]}^{f(\{i_1,\dots ,i_r\})}\cdot w_1$$

$$w_g=w_0\cdot [X_{i_1},\dots ,X_{i_r}{]}^{g(\{i_1,\dots ,i_r\})}\cdot w_2,$$

$$(w_f{)}_G(g_1,\dots ,g_d)=w_0(g_1,\dots ,g_d)\cdot [g_{i_1},\dots ,g_{i_r}{]}^{f(\{i_1,\dots ,i_r\})}$$

$$\ne w_0(g_1,\dots ,g_d)[g_{i_1},\dots ,g_{i_r}{]}^{g(\{i_1,\dots ,i_r\})}=(w_g{)}_G(g_1,\dots ,g_d),$$

$$w_{F_c(X_1,\dots ,X_d)}(X_1,\dots ,X_d)=\prod _{j=1}^{N_{d,c}}((X_{{\alpha }_j^{(d,c)}}{)}^{k_j}).$$

Full text

1 Introduction

1.1 Motivation and main results

In recent years, various authors have made contributions to the theory of word maps on groups; we refer interested readers to the survey article [2] for an overview of results and open problems of this theory.

Recall that a word is just an element $w$ of some free group $\operatorname{F}(X_{1},\ldots,X_{d})$ and that each such word gives rise to a word map $w_{G}:G^{d}\rightarrow G$ on every group $G$, induced by substitution. For a fixed finite group $G$, denote by $\Omega_{d}(G)$ the number of functions $G^{d}\rightarrow G$ that are of the form $w_{G}$ for some $w\in\operatorname{F}(X_{1},\ldots,X_{d})$. Moreover, set $\omega_{d}:=\log_{2}{\Omega_{d}(G)}$ (binary logarithm). The aim of this note is to show the following:

Theorem 1.1.1**.**

Let $G$ be a finite group, $c$ a non-negative integer.

If $G$ is nilpotent of class exactly $c$, then $\omega_{d}(G)=\Theta(d^{c})$ as $d\to\infty$. 2. 2.

If $G$ is not nilpotent, then $\omega_{d}(G)\geq 2^{d}-1$.

In particular, if $G$ is nilpotent of class $c$, then for some large enough constant $C=C(G)>0$, we have $\Omega_{d}(G)\leq C^{d^{c}}$ for all non-negative integers $d$, whereas $\Omega_{d}(G)$ grows doubly exponentially in $d$ if $G$ is not nilpotent. Note that the trivial upper bound $\Omega_{d}(G)\leq|G|^{|G|^{d}}$ is also doubly exponential in $d$ for fixed $G$. By Theorem 1.1.1, we also have the following quantitative characterization of nilpotency of finite groups:

Corollary 1.1.2**.**

A finite group $G$ is nilpotent (of class exactly $c$) if and only if the sequence $(\omega_{d}(G))_{d\geq 0}$ is of polynomial growth (of degree exactly $c$).

1.2 Notation

We denote by $\mathbb{N}$ the set of natural numbers (including [math]) and by $\mathbb{N}^{+}$ the set of positive integers. The set of all nonempty subsets of a set $X$ is denoted by $\mathcal{P}_{{}\not=\emptyset}(X)$. The exponent of a finite group $G$ is denoted by $\exp(G)$. As in the previous subsection, we denote by $\operatorname{F}(X_{1},\ldots,X_{d})$ the free group on the formal generators $X_{1},\ldots,X_{d}$, and we will denote by $\operatorname{F}_{c}(X_{1},\ldots,X_{d})$ the free nilpotent group of class exactly $c$ on the formal generators $X_{1},\ldots,X_{d}$.

1.3 Normal forms of elements in free nilpotent groups

For the reader’s convenience, we briefly recall a result on normal forms of elements in free nilpotent groups (based on a certain polycyclic decomposition of such groups) which we will need, see the blog post [3] or the textbook [1, Theorem 5.13A, p. 343, and its proof] for more details.

Fix $d,c\in\mathbb{N}$, and assign to each $i\in\{1,\ldots,d\}$ the weight $\operatorname{wt}(i):=1$. Consider formal iterated commutators of the numbers $1,\ldots,d$ (we will henceforth call them formal $d$-commutators) and define the weight of a formal $d$-commutator recursively via $\operatorname{wt}([\alpha,\beta]):=\operatorname{wt}(\alpha)+\operatorname{wt}(\beta)$. Moreover, for a formal $d$-commutator $\gamma$, define $X_{\gamma}$, a word in the variables $X_{1},\ldots,X_{d}$, recursively via $X_{[\alpha,\beta]}:=[X_{\alpha},X_{\beta}]=X_{\alpha}^{-1}X_{\beta}^{-1}X_{\alpha}X_{\beta}$.

Then there exists an explicitly constructible finite ordered tuple $(\alpha_{1}^{(d,c)},\ldots,\alpha_{N_{d,c}}^{(d,c)})$ of pairwise distinct formal $d$-commutators each of weight at most $c$ such that every element of $\operatorname{F}_{c}(X_{1},\ldots,X_{d})$ has a unique representation of the form $\prod_{j=1}^{N_{d,c}}{(X_{\alpha_{j}^{(d,c)}})^{k_{j}}}$ with $k_{j}\in\mathbb{Z}$.

For later use, we also note the following, which can be easily proved by induction on $c$:

Lemma 1.3.1**.**

For each $c\in\mathbb{N}$, there exists a polynomial $P_{c}(X)\in\mathbb{Z}[X]$ of degree $c$ such that for every $d\in\mathbb{N}$, the number of formal $d$-commutators of weight at most $c$ is exactly $P_{c}(d)$.∎

See also [1, Problems for Section 5.2, Problem 5] for a precise formula for the coefficients of $P_{c}(d)$.

2 Proof of Theorem 1.1.1

We begin by providing a lower bound on $\Omega_{d}(G)$ which will yield both statement (2) of Theorem 1.1.1 and half of statement (1). For this, we need some notation.

Notation 2.1**.**

Let $G$ be a group.

For $r\in\mathbb{N}^{+}$ and elements $g_{1},\ldots,g_{r}\in G$, we define their nested commutator $[g_{1},\ldots,g_{r}]$ recursively via $[g_{1}]:=g_{1}$ and $[g_{1},\ldots,g_{r+1}]:=[[g_{1},\ldots,g_{r}],g_{r+1}]$. 2. 2.

If $G$ is finite and $r\in\mathbb{N}^{+}$, then denote by $\exp_{r}(G)$ the least common multiple of the orders of the elements of $G$ of the form $[g_{1},\ldots,g_{r}]$, where the $g_{i}$ range over $G$.

Note that for fixed finite $G$, the sequence $(\exp_{r}(G))_{r\geq 1}$ is monotonically decreasing and that $\exp_{1}(G)=\exp(G)$.

Definition 2.2**.**

Let $G$ be a finite group, $d\in\mathbb{N}$.

A function $f:\mathcal{P}_{{}\not=\emptyset}(\{1,\ldots,d\})\rightarrow\mathbb{N}$ is called $G$-admissible if and only if for each $r\in\{1,\ldots,d\}$, we have $f(M)<\exp_{r}(G)$ for every $r$-element subset $M\subseteq\{1,\ldots,d\}$. 2. 2.

To each $G$-admissible function $f$, we assign a word $w_{f}(X_{1},\ldots,X_{d})$, defined as follows:

[TABLE]

We now show:

Proposition 2.3**.**

Let $G$ be a finite group, $d\in\mathbb{N}$.

Let $f,g$ be distinct $G$-admissible functions $\mathcal{P}_{{}\not=\emptyset}(\{1,\ldots,d\})\rightarrow\mathbb{N}$. Then the word maps $(w_{f})_{G}$ and $(w_{g})_{G}$ are distinct functions $G^{d}\rightarrow G$. 2. 2.

$\Omega_{d}(G)\geq\prod_{r=1}^{d}{\exp_{r}(G)^{{d\choose r}}}$.

Proof.

Statement (2) follows from statement (1), as the asserted lower bound is by definition just the number of $G$-admissible functions $\mathcal{P}_{{}\not=\emptyset}(\{1,\ldots,d\})\rightarrow\mathbb{N}$.

As for statement (1), let $r\in\mathbb{N}^{+}$ be minimal such that $f$ and $g$ disagree on some $r$-element subset of $\{1,\ldots,d\}$ and let $i_{1}<i_{2}<\cdots<i_{r}$ be such that $\{i_{1},\ldots,i_{r}\}$ is minimal with respect to the lexicographical ordering among all $r$-element subsets of $\{1,\ldots,d\}$ on which $f$ and $g$ have different values. Set $a:=f(\{i_{1},\ldots,i_{r}\})-g(\{i_{1},\ldots,i_{r}\})$ and note that $a\in\{1,\ldots,\exp_{r}(G)-1\}$. By definition of $\exp_{r}(G)$, this means that we can fix $g_{i_{1}},\ldots,g_{i_{r}}\in G$ such that

[TABLE]

Moreover, set $g_{i}:=1_{G}$ for $i\in\{1,\ldots,d\}\setminus\{i_{1},\ldots,i_{r}\}$.

We claim that $(w_{f})_{G}$ and $(w_{g})_{G}$ disagree on the argument $(g_{1},\ldots,g_{d})$. To see this, note that by choice of $(i_{1},\ldots,i_{r})$, we have

[TABLE]

and

[TABLE]

where $w_{0},w_{1},w_{2}\in\operatorname{F}(X_{1},\ldots,X_{d})$ and $w_{1}$, $w_{2}$ are products of nested commutators of length at least $r$ distinct from $[X_{i_{1}},\ldots,X_{i_{r}}]$, so that when substituting $g_{i}$ for $X_{i}$, at least one of the entries of each such nested commutator will be $1_{G}$, forcing it to be trivial. Therefore,

[TABLE]

as required. ∎

Corollary 2.4**.**

Let $G$ be a finite group, $c\in\mathbb{N}$, and assume that the $(c+1)$-th term in the lower central series of $G$ is nontrivial. Then for every $d\in\mathbb{N}$, $\omega_{d}(G)\geq\sum_{r=1}^{c}{{d\choose r}}$.

Proof.

By assumption, $\exp_{r}(G)\geq 2$ for $r=1,\ldots,c$, so the result follows immediately from Proposition 2.3(2). ∎

In particular, $d^{c}=\mathcal{O}(\omega_{d}(G))$ whenever $G$ is nilpotent of class exactly $c$ (which is “half of Theorem 1.1.1(1)”) and if $G$ is not nilpotent, we can choose $c:=d$ in Corollary 2.4 and get Theorem 1.1.1(2) using that $\sum_{r=1}^{d}{{d\choose r}}=2^{d}-1$. The second half of Theorem 1.1.1(1) is provided by the following:

Lemma 2.5**.**

Let $c\in\mathbb{N}$, and let $P_{c}(X)\in\mathbb{Z}[X]$ be as in Lemma 1.3.1. Then for every finite nilpotent group $G$ of class exactly $c$, we have $\omega_{d}(G)\leq P_{c}(c)\cdot\log_{2}(\exp(G))$.

Proof.

We show the equivalent $\Omega_{d}(G)\leq\exp(G)^{P_{c}(c)}$. Let $w\in\operatorname{F}(X_{1},\ldots,X_{d})$. Then the remarks in Subsection 1.3 yield that

[TABLE]

for suitable integers $k_{j}$. As $\operatorname{F}_{c}(X_{1},\ldots,X_{d})$ is the free object in the category of nilpotent groups of class at most $c$, this relation among its generators translates to an identity in all nilpotent groups of class at most $c$, yielding in particular that $w_{G}=(\prod_{j=1}^{N_{d,c}}((X_{\alpha_{j}^{(d,c)}})^{k_{j}}))_{G}$. Hence each word map on $G$ is induced by a word of the form $\prod_{j=1}^{N_{d,c}}((X_{\alpha_{j}^{(d,c)}})^{k_{j}})$ for suitable integers $k_{j}$, which we may w.l.o.g. assume to be from the finite range $\{0,\ldots,\exp(G)-1\}$. Therefore, $\Omega_{d}(G)\leq\exp(G)^{N_{d,c}}\leq\exp(G)^{P_{c}(c)}$, as required. ∎