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 \(\imath_G\) 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.Mathematics Subject Classifications: 05E16, 20D60, 05B15Keywords: Sequenceable groups, Latin squares, harmonious groups, complete mappings