Strong complete mappings for 3-groups

Abstract

A strong complete mapping for a group G is a bijection φ:G→G such that the maps x↦xφ(x) and x↦x−1φ(x) are also bijections. Groups admitting a strong complete mapping are important to the study of orthogonality problems for Latin squares and group sequencings, among other applications. A.B. Evans [6] showed that a finite abelian group admits a strong complete mapping if and only if both its 2-Sylow subgroup and its 3-Sylow subgroup are either trivial or noncyclic. Nilpotent groups resemble abelian groups in that they also possess the property of being the direct product of their Sylow subgroups; therefore, it is natural to begin consideration of the nonabelian case by asking which nilpotent groups admit strong complete mappings. As the function x↦x2 furnishes a strong complete mapping for finite groups of order relatively prime to 6, we need only consider 2-groups and 3-groups. As a step in this direction, we prove that every noncyclic 3-group admits a strong complete mapping, except possibly those in the infinite family Lr=〈a,b|a3r−1=b3=1,bab−1=a1+3r−2〉, r≥4.