Partitions of groups and complete mappings

Richard J. Friedlander, Basil Gordon and Peter Tannenbaum

Vol. 92 (1981), No. 2, 283–293

DOI: 10.2140/pjm.1981.92.283

Abstract

Let G be an abelian group of order n and let k be a divisor of n− 1. We wish to determine whether there exists a complete mapping of G which fixes the identity element and permutes the remaining elements as a product of disjoint k-cycles. We conjecture that if G has trivial or noncyclic Sylow 2-subgroup then such a mapping exists for every divisor k of n− 1. Several special cases of the conjecture are proved in this paper. We also prove that a necessary condition for the existence of such a map holds for every k when G is cyclic.

Mathematical Subject Classification 2000

Primary: 20K99

Secondary: 20G99

Milestones

Received: 12 September 1979

Revised: 8 April 1980

Published: 1 February 1981