On the triplet-divisions of the set \(\{1,2,3,\cdots,p-2\}\) (Q1272706)

The author shows that the numbers in the set \(\{1, 2, \ldots, p-2\}\) with \(p\geq 5\) can be divided into disjoint triplets \(\langle a, b, c\rangle\) such that \(abc\equiv 1 \pmod{p}\). This is a generalization of the doublet-division of the numbers in the set \(\{1, 2, \ldots, p-1\}\) into disjoint pairs \(\langle a, a^\ast \rangle\) such that \(aa^\ast\equiv 1 \pmod{p}\). The triplet-division bases on the properties of the map \(F: \{1, 2, \ldots, p-2\}\to \{1, 2, \ldots, p-2\}, F(a)=p-1-a^\ast\), such as \(aF(a)F^2(a)\equiv 1 \pmod{p}\).