Abstract
In the paper, we develop further the properties of Schur rings over infinite groups, with particular emphasis on the virtually cyclic group \(\mathcal {Z}\times \mathcal {Z}_p\), where p is a prime. We provide structure theorems for primitive sets in these Schur rings. In the case of Fermat and safe primes, a complete classification theorem is proven, which states that all Schur rings over \(\mathcal {Z}\times \mathcal {Z}_p\) are traditional. We also draw analogs between Schur rings over \(\mathcal {Z}\times \mathcal {Z}_p\) and partitions of difference sets over \(\mathcal {Z}_p\).
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Notes
OEIS sequence A019434.
OEIS sequence A005385. Note that the associated prime q to the safe prime \(p=2q+1\) is called a Sophie Germain prime Safe primes received their name because of their historical usage in cryptography.
These omitted proofs can be found in [1].
We say that a Schur ring \(\mathfrak {S}\) is primitive if it has no nontrivial, proper \(\mathfrak {S}\)-subgroups (this definition, while equivalent, is not the definition originally used by Wielandt or Scott). The primitive Schur rings have been studied extensively, particularly because of their applications to permutation groups. In [6], Scott uses 13.8.2 and 13.8.3 to rule out the existence of primitive Schur rings over groups of small order in a way analogous to how Sylow theorems are used to show the non-existence of simple groups of small order.
A difference set D is a subset of a group G such that every non-identity element in G appears in \(DD^*\) with the same multiplicity. When G is finite, this is equivalent to \({\overline{D}}\,{\overline{D}}^* = n+\lambda {\overline{G}}\). Thus, difference sets guarantee the greatest equity among the multiplicities of group elements in the product \(DD^*\). As the multiplicity of any element in the product \(AB^*\) ranges between 0 and |A|, a set containing a tycoon represents the greatest possible inequity between multiplicities of group elements. If multiplicities are replaced by wealth for the sake of analogy, the curious name tycoon is then explained.
In the case that \([\mathcal {Z}:H]=\infty \), that is, \(H=1\), set \(K:=1\) and the above result and proof would remain valid.
Note that this argument does not require p to be prime and hence applies to all positive integers n.
If \(\mathfrak {S}=F[G]^\mathcal {H}\) and \(\tau \in \mathop {\textrm{Aut}}\limits (G)\), then \(\tau (\mathfrak {S}) = F[G]^{\tau \mathcal {H}\tau ^{-1}}\).
References
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Acknowledgements
All calculations made in preparation of this paper were made using MAGMA [2]. The authors are grateful to Stephen Humphries for useful conversations and to the anonymous referees for their most helpful comments.
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Bastian, N., Misseldine, A. On Schur rings over infinite groups III. J Algebr Comb 58, 1261–1276 (2023). https://doi.org/10.1007/s10801-023-01273-z
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DOI: https://doi.org/10.1007/s10801-023-01273-z