Doubly Sequenceable Groups

TL;DR

This paper introduces the concept of doubly sequenceable groups, characterizes their properties, and proves that certain group products are doubly sequenceable, expanding understanding of sequenceability in group theory.

Contribution

It defines doubly sequenceable groups and proves that the product of an odd or sequenceable group with an abelian group is doubly sequenceable.

Findings

Groups with certain properties are doubly sequenceable.

Product of an odd or sequenceable group with an abelian group is doubly sequenceable.

Doubly sequenceable groups include all abelian, odd, sequenceable, R-sequenceable, or terraceable groups.

Abstract

Given a sequence ${\bf g}: g_0,\ldots, g_{m}$, in a finite group $G$ with $g_0=1_G$, let ${\bf \bar g}: \bar g_0,\ldots, \bar g_{m}$, be the sequence defined by $\bar g_0=1_G$ and $\bar g_i=g_{i-1}^{-1}g_i$ for $1\leq i \leq m$. We say that $G$ is doubly sequenceable if there exists a sequence ${\bf g}$ in $G$ such that every element of $G$ appears exactly twice in each of ${\bf g}$ and ${\bf \bar g}$. If a group $G$ is abelian, odd, sequenceable, R-sequenceable, or terraceable, then $G$ is doubly sequenceable. In this paper, we show that if $N$ is an odd or sequenceable group and $H$ is an abelian group, then $N \times H$ is doubly sequenceable.