Abstract
In 1993, Muzychuk [Citation23] 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 [Citation24] 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.
2010 Mathematics Subject Classification:
ACKNOWLEDGMENTS
Most of the results in this article were included as part of the author's Master's thesis at Brigham Young University, under the supervision of Stephen P. Humphries [Citation17]. The author expresses gratitude to an anonymous reviewer who provided a simplified proof of Theorem 5.1.
Notes
Communicated by S. Sehgal.