Graceful Ordering of Abelian Groups

Mohammad Javaheri

Published 2022-08-30Version 1

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 prove that if $G$ is abelian then there exists a sequence ${\bf g}$ such that each element of $G$ appears exactly twice in each of ${\bf g}$ and ${\bf \bar g}$.