Harmonious sequences in groups with a unique involution

TL;DR

This paper explores combinatorial properties of finite abelian groups related to harmonious sequences, establishing new existence results for specific permutations and sequences with particular sum properties, and applying these to Latin square transversals.

Contribution

It proves the existence of special permutations and sequences in abelian groups with a unique involution, advancing understanding of sequenceability and harmonious sequences.

Findings

Existence of permutations with consecutive sums forming a permutation of the group minus the involution.

Existence of sequences containing each non-identity element twice with sum sequences also containing each non-identity element twice.

Application of results to the existence of transversals in Latin squares.

Abstract

We study several combinatorial properties of finite groups that are related to the notions of sequenceability, R-sequenceability, and harmonious sequences. In particular, we show that in every abelian group $G$ with a unique involution $\imath_G$ there exists a permutation $g_0,\ldots, g_{m}$ of elements of $G \backslash \{\imath_G\}$ such that the consecutive sums $g_0+g_1, g_1+g_2,\ldots, g_{m}+g_0$ also form a permutation of elements of $G\backslash \{\imath_G\}$. We also show that in every abelian group of order at least 4 there exists a sequence containing each non-identity element of $G$ exactly twice such that the consecutive sums also contain each non-identity element of $G$ twice. We apply several results to the existence of transversals in Latin squares.