Strong complete mappings for 2-groups

Abstract

A strong complete mapping for a group G is a bijection $\phi :G\to G$ such that the maps $x\mapsto x\phi \left(x\right)$ and $x\mapsto x^{-1}\phi \left(x\right)$ 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. In previous work we showed that a finite 3-group that contains no cyclic subgroup of index 3 is strongly admissible. In this article, we employ a substantially different strategy to show that a finite 2-group that contains no cyclic subgroup of index 4 is strongly admissible.