Sequencings in Semidirect Products via the Polynomial Method

TL;DR

This paper explores sequenceability properties in semidirect products of groups, employing the polynomial method to establish conditions under which subsets can be linearly or weakly sequenced, with applications to dihedral and non-abelian groups.

Contribution

It introduces new results on sequenceability in semidirect products, especially dihedral and non-abelian groups, using the polynomial method to identify when subsets are linearly or weakly sequenceable.

Findings

Every subset of size up to 12 in dihedral groups has a linear sequencing under certain conditions.

Subsets of size 5 to 10 in non-abelian groups of order 3 times a prime are linearly sequenceable unless the last partial sum is the identity.

Large subsets of groups with order pe are t-weakly sequenceable when specific size and prime conditions are met.

Abstract

The partial sums of a sequence ${\mathbf x} = x_1, x_2, \ldots, x_k$ of distinct non-identity elements of a group $(G,\cdot)$ are $s_0 = id_G$ and $s_j = \prod_{i=1}^j x_i$ for $0 < j \leq k$. If the partial sums are all different then ${\mathbf x}$ is a linear sequencing and if the partial sums are all different when $|i-j| \leq t$ then ${\mathbf x}$ is a $t$-weak sequencing. We investigate these notions of sequenceability in semidirect products using the polynomial method. We show that every subset of order $k$ of the non-identity elements of the dihedral group of order $2m$ has a linear sequencing when $k \leq 12$ and either $m>3$ is prime or every prime factor of $m$ is larger than $k!$, unless $s_k$ is unavoidably the identity; that every subset of order $k$ of a non-abelian group of order three times a prime has a linear sequencing when $5 < k \leq 10$, unless $s_k$ is unavoidably the identity; and that if the order of a group is $pe$ then all sufficiently large subsets of the non-identity elements are $t$-weakly sequenceable when $p>3$ is prime, $e \leq 3$ and $t \leq 6$.