On Schur 2-groups
Mikhail Muzychuk, Ilya Ponomarenko

TL;DR
This paper classifies Schur 2-groups, showing that certain abelian groups are Schur and that large non-abelian Schur 2-groups are dihedral, with detailed analysis of Schur rings of small rank.
Contribution
It completes the classification of abelian 2-groups as Schur and characterizes non-abelian Schur 2-groups of order greater than 32 as dihedral, also analyzing Schur rings of small rank.
Findings
Z_2 × Z_{2^n} is Schur
Non-abelian Schur 2-groups > 32 are dihedral
Unique obstacle is a hypothetical S-ring of rank 5
Abstract
A finite group is called a Schur group, if any Schur ring over is the transitivity module of a point stabilizer in a subgroup of that contains all right translations. We complete a classification of abelian -groups by proving that the group is Schur. We also prove that any non-abelian Schur -group of order larger than is dihedral (the Schur -groups of smaller orders are known). Finally, in the dihedral case, we study Schur rings of rank at most , and show that the unique obstacle here is a hypothetical S-ring of rank associated with a divisible difference set.
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Taxonomy
TopicsFinite Group Theory Research · Mathematics and Applications · Advanced Topics in Algebra
