On Schur 2-Groups

Abstract

A finite group G is called a Schur group if every Schur ring over G is the transitivity module of a point stabilizer in a subgroup of Sym(G) that contains all permutations induced by the right multiplications in G. It is proved that the group \( {\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_{2^n} \) is Schur, which completes the classification of Abelian Schur 2-groups. It is also proved that any non-Abelian Schur 2-group of order larger than 32 is dihedral (the Schur 2-groups of smaller orders are known). Finally, the Schur rings over a dihedral 2-group G are studied. It turns out that among such rings of rank at most 5, the only obstacle for G to be a Schur group is a hypothetical ring of rank 5 associated with a divisible difference set.