Automorphism Groups of Schur Rings

In 1993, Muzychuk [18] showed that the rational Schur rings over a cyclic group Zn are in one-to-one correspondence with sublattices of the divisor lattice of n, or equivalently, with sublattices of the lattice of subgroups of Zn. 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 [19] 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. Schur rings (or S-rings), first developed by I. Schur in [20] in 1933 as a tool for investigating permutation groups, have subsequently found several applications in graph theory and algebraic combinatorics. For a survey of the recent developments in S-rings and their applications, see [17]. All of the definitions and elementary results that we need will be reviewed in §1. A great deal of effort has been focused on obtaining a classification of S-rings over cyclic groups, which was achieved in [13, 14]. After the successful classification of S-rings over cyclic groups, a logical next step has been to seek classifications for S-rings over broader families of groups. This, however, appears to be a very difficult problem even in the case, for instance, of abelian p-groups. One approach is to restrict attention to special types of S-rings, in particular rational S-rings, in the hope that understanding these may shed light on the general problem. In §2 we describe a general construction which, for each sublattice of the lattice of characteristic subgroups of a group G, produces a corresponding rational S-ring over G. This construction generalizes the one in [18] used to classify rational S-rings over cyclic groups. In §3 we review the structure of the lattice of characteristic subgroups over abelian p-groups. This enables us to explicitly describe the construction of many rational S-rings over such groups, which we will need for our main result in §4.