## 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 ℤ2×ℤ2n 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.

Original language | English |
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Pages (from-to) | 565-594 |

Number of pages | 30 |

Journal | Journal of Mathematical Sciences |

Volume | 219 |

Issue number | 4 |

DOIs | |

State | Published - 1 Dec 2016 |

Externally published | Yes |

## ASJC Scopus subject areas

- Statistics and Probability
- Mathematics (all)
- Applied Mathematics