Automorphism Groups of Schur Rings

Abstract

In 1993, Muzychuk \cite{muzychuk} 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, we show that 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 \cite{muzychuk2} 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.