On Schur Rings over Infinite Groups

Abstract

In the paper, we develop further the properties of Schur rings over infinite groups, with particular emphasis on the virtually cyclic group Z × Zp. 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 Z × Zp are traditional. We also draw analogs between Schur rings over Z × Zp and partitions of difference sets over Zp.

Schur Rings over Z \times Z_3

“…The following proposition was proved by Wielandt in [11,Proposition 22.3]. The general case appeared as [1,Propostion 2.4].…”

“…The following two propositions, which were first proved by Wielandt in [11] and were generalized in [1], tell us how to generate S-subgroups in any S-ring. When we have normal S-subgroup, we can construct S-ring over the factor group as shown in the following lemma.…”

“…In addition, we define C * = {g −1 : g ∈ C}. A partition P of G is said to be finite support provided any C ∈ P is a finite subset of G. Our notation for Schur ring is taken from [1] and [8]. For an S-ring A over G, the associated partition P is denoted D(A) and each element in D(S) is called a basic set of A.…”

“…Actually, it is proved that the only Schur ring over Z are discrete and symmetric Schur rings; see [1,Theorem 3.3]. The symmetric Schur ring over Z is denoted…”

“…Rcently, Bastian et al [1] study the Schur rings over infinite groups. Schur rings over the integers, over a virtually infinite cyclic group, and over torsion-free locally cyclic groups are classified.…”