On Schur 2-groups

Ilya Ponomarenko
St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences

Misha Muzychuk
Netanya Academic College, Netanya, Israel

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Minisymposium: ASSOCIATION SCHEMES

Content: A finite group $G$ is called a Schur group, if any Schur ring over~$G$ is the transitivity module of a point stabilizer in a subgroup of Sym($G$) that contains all right translations. We complete a classification of abelian 2-groups by proving that the group $Z_2\times Z_{2^n}$ is Schur. We also prove that any non-abelian Schur 2-group of order larger than 32 is dihedral (the Schur 2-groups of smaller orders are known). Finally, in the dihedral case, we study Schur rings of rank at most 5, and show that the unique obstacle here is a hypothetical S-ring of rank 5 associated with a divisible difference set.

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