Integral table algebras and Bose-Mesner algebras with a faithful nonreal element of degree three
Theorem 1. Let \((A,{\mathbf B})\) be a standard integral table algebra without nontrivial linear elements. If there exist a faithful nonreal element \(a\in {\mathbf B}\) of degree 3 and \(a\bar a=3\cdot {\mathbf 1}+b\) for some \(b\in \mathbf B\) with \(|b|=6\), then \({\mathbf B}\) is isomorphic \({\mathbf G}_n\) for some \(n\geq 3\) or \({\mathbf O}(S_3,{\mathbf Z}_m\times {\mathbf Z}_m)\) for some \(m\geq 4\). The combination of the main theorem of \textit{Z. Arad, E. Arisha, M. Fisman} and \textit{M. Muzychuk} [J. Algebra 231, No. 2, 473-483 (2000; review above)], the theorem B of \textit{H. I. Blau} (2000) and Theorem 1 yields Theorem 2. Let \((A,{\mathbf B})\) be a standard integral table algebra without nontrivial linear elements. If there exist a faithful nonreal element \(a\in {\mathbf B}\) of degree 3, then \({\mathbf B}\) is isomorphic to one of the following: a wreath product of \({\mathbf V}\) and \({\mathbf Z}_m\) for some \(m\geq 3\); \({\mathbf T}_n(3)\) for some \(n\geq 1\); \({\mathbf G}_n\) for some \(n\geq 3\); \({\mathbf O}({\mathbf Z}_3,H)\) for some abelian group \(H\) of rank at most 2; or \({\mathbf O}(S_3,{\mathbf Z}_m\times {\mathbf Z}_m)\) for some \(m\geq 3\).
scientific article; zbMATH DE number 1526422