Harmonious sequences in groups with a unique involution

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 ${\mathrm{ı}}_G$ there exists a permutation $g_0,\dots ,g_m$ of elements of $G\mathrm{\backslash }\{{\mathrm{ı}}_G\}$ such that the consecutive sums $g_0+g_1,g_1+g_2,\dots ,g_m+g_0$ also form a permutation of elements of $G\mathrm{\backslash }\{{\mathrm{ı}}_G\}$. 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.