Translation nets and fixed-point-free group automorphisms
The authors discuss the equivalence of three mathematical objects: translation nets (i.e. nets with a regular automorphism group which preserves parallel classes), sets of MOLS with a regular group of strong automorphisms, and partial congruence partitions of a group (i.e. sets of subgroups of the same order \(n\), if \(n^ 2\) is the order of the group, such that any two subgroups have a trivial intersection). They review known results (and give good references), provide a construction which shows the equivalence above, with the help of a fixed- point-free set of automorphism of a group. Bounds are proven for the maximum size of a set of MOLS. In the special case of an abelian p-group H, the exact values are given for the maximum size of a set of MOLS based on \(H\) (and constructed by Mann's automorphism method), for the number of parallel classes of a translation net (with \(H\times H\) as a translation group) and for the maximal degree (number of subgroups) of a partial congruence partition of \(H\times H\). Improvements are also provided of the known bounds for non-abelian \(p\)-groups. Some applications are shown to partial difference sets, strongly regular graphs and design of experiments.
scientific article; zbMATH DE number 4179386
scientific article