[Paper Review] Extremal product-one free sequences in $C_q times_s C_m$

Fabio Enrique Brochero Martínez, Sávio Ribas|arXiv (Cornell University)|Oct 31, 2016

Limits and Structures in Graph Theory3 citations

TL;DR

This paper characterizes all extremal product-one free sequences in the metacyclic group $C_q \rtimes_s C_m$, where $q$ is prime, $m \geq 2$ divides $q-1$, and $\mathrm{ord}_q(s) = m$. It proves that any such sequence of length $q + m - 2$ must consist of elements from a single coset of the normal subgroup $C_q$, with exponents forming a specific arithmetic structure, and that no such sequence can avoid containing a subsequence whose product is the identity unless it is highly structured, thus resolving the inverse problem for this class of non-abelian groups.

ABSTRACT

Let $G$ be a finite group, written multiplicatively. The Davenport constant of $G$ is the smallest positive integer $d$ such that every sequence of $G$ with $d$ elements has a non-empty subsequence with product $1$. Let $C_n \simeq \mathbb Z_n$ be the cyclic group of order $n$. Bass (2007) showed that the Davenport constant of the metacyclic group $C_q times_s C_m$, where $q$ is a prime number and $ ext{ord}_q(s) = m \ge 2$, is $m+q-1$. In this paper, we explicit the form of all sequences $S$ of $C_q times_s C_m$, with $q+m-2$ elements, that are free of product-$1$ subsequences.

Motivation & Objective

• To determine the structure of all maximal sequences in $C_q \rtimes_s C_m$ that are free of non-empty subsequences with product 1.
• To resolve the inverse problem for the Davenport constant in the non-abelian metacyclic group $C_q \rtimes_s C_m$, where $q$ is prime and $\mathrm{ord}_q(s) = m \geq 2$.
• To extend the understanding of extremal sequences beyond abelian groups, particularly in non-abelian groups of rank two.
• To show that such extremal sequences must be highly structured, resembling the cyclic case in their repetition and distribution across cosets.

Proposed method

• The authors analyze sequences of length $q + m - 2 = D(G) - 1$ in $G = C_q \rtimes_s C_m$, focusing on their distribution across cosets of the normal subgroup $C_q \triangleleft G$.
• They use group-theoretic techniques, including the action of $x^{i_0}$ on $y$-exponents via conjugation: $x^{i_0} y^a x^{-i_0} = y^{a s^{i_0}}$, to analyze the structure of sequences in the coset $x^{i_0} C_q$.
• They define sets $X$ and $Y$ of $\mathbb{Z}_q$-exponents derived from subsequence products and use additive combinatorics to analyze whether $0 \in X + Y$, which would imply a product-one subsequence.
• They apply results on quadratic residues modulo $q$, particularly the fact that $-1$ is a quadratic residue iff $q \equiv 1 \pmod{4}$, to derive contradictions when assuming no product-one subsequence exists.
• They use permutation-based arguments on exponent sums to construct potential product-one subsequences, showing that any sufficiently large or diverse sequence must contain such a subsequence.
• They handle exceptional cases like $(m,q) = (2,3)$ and $(4,5)$ separately, using direct verification and structural analysis.

Experimental results

Research questions

• RQ1 What is the complete structure of all sequences of length $q + m - 2$ in $C_q \rtimes_s C_m$ that are free of non-empty subsequences with product 1?
• RQ2 Can such extremal sequences be characterized in terms of their distribution across cosets of $C_q$?
• RQ3 Does the inverse Davenport constant problem for $C_q \rtimes_s C_m$ yield sequences with a structure analogous to the cyclic case, where all elements are equal?
• RQ4 Under what conditions on $q$, $m$, and $s$ does a sequence of length $q + m - 2$ avoid having a product-one subsequence?
• RQ5 How does the non-abelian structure of $C_q \rtimes_s C_m$ affect the existence and form of extremal product-one free sequences?

Key findings

• All extremal product-one free sequences in $C_q \rtimes_s C_m$ of length $q + m - 2$ are contained within a single coset $x^{i_0} C_q$ for some $i_0 \in \{1, \dots, m-1\}$.
• Such sequences must have all $y$-exponents forming a set $A \subset \mathbb{Z}_q$ such that the sumset $A + s^{i_0} A$ avoids 0 modulo $q$, under specific structural constraints.
• If $q \equiv 1 \pmod{4}$ and $q \geq 13$, then any sequence with $m$ distinct elements in a coset must contain a product-one subsequence, due to the existence of quadratic residue decompositions.
• If $q \equiv 3 \pmod{4}$, the impossibility of $-1$ being a quadratic residue leads to a contradiction if $X + Y = \mathbb{Z}_q^*$ and $-\alpha \in X$ for $\alpha \in X$, forcing the existence of a product-one subsequence.
• The only possible extremal sequences are those with at most $m-1$ distinct elements in a coset, and when $m = q-1$, the sequence must be composed of $m-1$ copies of a single element to avoid forming a product-one subsequence.
• The exceptional case $(m,q) = (2,3)$ is excluded due to the group being isomorphic to $S_3$, and the case $(4,5)$ is handled separately via direct computation.