On Schur 2-groups

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 $G$ 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 $\mZ_2\times\mZ_{2^n}$ 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.