Groups with anti-symmetric mappings (Q1901967)

We call a permutation \(\phi\) from a group to itself anti-symmetric if \(\phi(x)y = \phi(y)x\) implies that \(x = y\). Any anti-symmetric mapping of a group \(G\) can be used to assign a check digit to any string of characters from \(G\) so that all single errors and all transposition errors of adjacent characters are detected. When \(G\) is Abelian, anti-symmetric mappings are equivalent to the much studied complete mappings. In this paper we determine large classes of important groups that have anti-symmetric mappings and classes that do not. In particular, all finite simple groups except \(Z_2\) have anti-symmetric mappings. We also establish some general criteria for the existence of anti-symmetric mappings.