Antisymmetric mappings for finite solvable groups

A bijection phi from a group G to itself is called an antisymmetric mapping if for all g, h is an element of G with g not equal h: g phi(h)not equal h phi(g). It has been conjectured by J. A. Gallian and M. D. Mullin [3] that every non-abelian group possesses an antisymmetric mapping. The aim of this note is to supply a proof of this conjecture in the case of finite non-abelian solvable groups. Constructions of antisymmetric mappings are given explicitly for a number of solvable groups. Principally, these constructions allow a recursive construction of an antisymmetric mapping for every non-abelian solvable group.