On a huge family of non-schurian Schur rings

TL;DR

This paper constructs a large family of non-schurian Schur rings over elementary abelian groups of even rank, expanding on Wielandt's classical example and demonstrating their existence beyond previously known cases.

Contribution

It generalizes Wielandt's example by constructing a vast family of non-schurian Schur rings over elementary abelian groups of even rank.

Findings

Existence of large families of non-schurian Schur rings.

Extension of Wielandt's classical example to higher ranks.

Demonstration that non-schurian Schur rings are abundant in certain group classes.

Abstract

In his famous monograph on permutation groups, H.~Wielandt gives an example of a Schur ring over an elementary abelian group of order $p^2$ ($p>3$ is a prime), which is non-schurian, that is, it is the transitivity module of no permutation group. Generalizing this example, we construct a huge family of non-schurian Schur rings over elementary abelian groups of even rank.