The Strong Admissibility of Finite Groups: An Update

For a finite group \( G \), a bijection \( \theta: G \to G \) is a \emph{strong complete mapping} if the mappings \( g \mapsto g\theta(g) \) and \( g \mapsto g^{-1}\theta(g) \) are both bijections. A group is \emph{strongly admissible} if it admits strong complete mappings. Strong complete mappings have several combinatorial applications. There exists a Latin square orthogonal to both the multiplication table of a finite group \( G \) and its normal multiplication table if and only if \( G \) is strongly admissible. The problem of characterizing strongly admissible groups is far from settled. In this paper, we will update progress towards its resolution. In particular, we will present several infinite classes of strongly admissible dihedral and quaternion groups and determine all strongly admissible groups of order at most 31.