Antisymmetric mappings for finite solvable groups

摘要

A bijection o from a group G to itself is called an antisymmetric mapping if for all g,h in G with g =/ h g, o (h) = h, o(g) .It has been conjectured by J. A. Gallian and M. D. Mullin [15] 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.