On the harmoniousness of dicyclic groups (Q687129)

A group of order \(m\) is called harmonious if its elements can be listed in a sequence \(g_ 1,\dots,g_ m\) such that \(g_ 1 g_ 2\), \(g_ 2 g_ 3,\dots,g_{m-1}g_ m\), \(g_ mg_ 1\) are all distinct. In this paper, the author proves the following theorem: The dicyclic group \(Q_{2n}\) is harmonious if \(n \equiv 0\bmod 4\) or \(n\equiv 0\bmod 6\). The dicyclic group \(Q_{2n}\) of order \(4n\) is defined by \(Q_{2n}=\langle a,b : a^{2n}=1\), \(b^ 2=a^ n\), \(ab=ba^{- 1}\rangle\).