On $R$-sequenceability of odd ordered groups

Abstract

We study the $R$-sequenceability of finite groups of odd order. Building on the classical theory of $R^{\ast }$-sequences and orthomorphisms, we explore two tools: the notion of $R^{\ast \ast }$-sequenceability, a strengthening of $R^{\ast }$-sequenceability tailored for inductive arguments over normal subgroups with cyclic quotients, and the \textit{odd cycle index} $\ddot{I}„(G)$, which measures how many orthomorphisms are required to generate a full cycle together with an involution. Our main result is a Quotient-Normal Gadget theorem, which shows that if $G$ has a normal subgroup $N$ such that $G/N$ is $R^{\ast \ast }$-sequenceable and $\ddot{I}„(N)\le |G/N|-3$, then $G$ itself is $R^{\ast \ast }$-sequenceable. We prove that $\ddot{I}„(G)=2$ for cyclic groups of order coprime with $3$, and establish an inductive bound $\ddot{I}„(G)\le \max \{\ddot{I}„(N),\ddot{I}„(G/N)\}$ for odd ordered groups with a normal subgroup $N$. As consequences, we show that every group whose order is coprime with $30$ is $R$-sequenceable, and that every nilpotent group whose order is coprime with $6$ and not a power of $5$ is $R$-sequenceable. These results extend prior work on abelian groups to broad families of non-abelian groups.