On $R$-sequenceability of odd ordered groups

We study the $R$-sequenceability of finite groups of odd order.

Building on the classical theory of $R^*$-sequences and orthomorphisms, we explore two tools: the notion of $R^{**}$-sequenceability, a strengthening of $R^*$-sequenceability tailored for inductive arguments over normal subgroups with cyclic quotients, and the \textit{odd cycle index} $\tau(G)$, 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 $R^{**}$-sequenceable and $\tau(N) \leq |G/N| - 3$, then $G$ itself is $R^{**}$-sequenceable.

We prove that $\tau(G) = 2$ for cyclic groups of order coprime with $3$, and establish an inductive bound $\tau(G) \leq \max\{\tau(N), \tau(G/N)\}$ 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.