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 G there exists a permutation g₀,…, gₘ of elements of G \ such that the consecutive sums g₀+g₁, g₁+g₂,…, gₘ+g₀ also form a permutation of elements of 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.