Automorphism Groups of Schur Rings

Abstract

In 1993, Muzychuk [23] 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] 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.

Key Words:

• Automorphism groups
• Characteristic subgroups
• Schur rings

2010 Mathematics Subject Classification:

• 16S34

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 [17]. The author expresses gratitude to an anonymous reviewer who provided a simplified proof of Theorem 5.1.

Notes

Communicated by S. Sehgal.