Automorphism Groups of Schur Rings

In 1993, Muzychuk 23 Muzychuk , Mikhail E. ( 1993 ). The structure of rational Schur rings over cyclic groups . European Journal of Combinatorics 14 : 479 – 490 .Crossref], Web of Science ®] , Google Scholar]] showed that the rational Schur rings over a cyclic group Z _n are in one-to-one correspondence with sublattices of the divisor lattice of n, or equivalently, with sublattices of the lattice of subgroups of Z _n . This can easily be extended to show that for any finite group G, sublattices of the lattice of characteristic subgroups of G give rise to rational Schur rings over G in a natural way. Our main result is that any finite group may be represented as the (algebraic) automorphism group of such a rational Schur ring over an abelian p-group, for any odd prime p. In contrast, over a cyclic group the automorphism group of any Schur ring is abelian. We also prove a converse to the well-known result of Muzychuk 24 Muzychuk , Mikhail E. ( 1994 ). On the structure of basic sets of Schur rings over cyclic groups . Journal of Algebra 169 : 655 – 678 .Crossref], Web of Science ®] , Google Scholar]] that two Schur rings over a cyclic group are isomorphic if and only if they coincide; namely, we show that over a group which is not cyclic, there always exist distinct isomorphic Schur rings.