Local nearrings with generalized quaternion multiplicative group

Abstract

A not necessarily zero-symmetric nearring R with identity is called local if the set of all non-invertible elements of R forms a subgroup of its additive group. The local nearrings whose multiplicative group is generalized quaternion are described. In particular, it is proved that their additive groups are abelian of

Classification of the affine structures of a generalized quaternion group of order ⩾32{\geqslant 32}

“…Factoring out this ideal turns every quaternion brace into a brace with a dihedral adjoint group. By induction, this implies that all subgroups of ha 4 i are brace ideals, while the Frattini subgroup ha 2 i of the adjoint group is still an additive subgroup (Proposition 1), reproving the results of Sysak et al [1,17] in a brace-theoretic manner. It turns out that the subgroup ha 2 i need not be a submodule under the adjoint group.…”

“…If true, this would provide an infinite sequence of groups with increasing order for which the number of affine structures stabilizes at a certain order. Some evidence for this phenomenon is given by papers of Sysak et al [1,17] which imply that the additive group of a quaternion brace must have a cyclic subgroup of index 4.…”

Finite local nearrings

“…Сисак та Дi Термiнi в [51] показали, що якщо R є локальним майже-кiльцем (не обов'язково нульсиметричним), мультиплiкативна група якого є узагальненою групою кватернiонiв, то мультиплiкативна група є або групою кватернiонiв порядку 8, або узагальненою групою кватернiонiв порядку 16 з абелевими адитивними групами таких типiв: Наступним природним кроком є дослiдження майже-полiв та локальних майже-кiлець, мультиплiкативна група яких є групою Мiллера -Морено, тобто мiнiмальною неабелевою. Майже-поля iз зазначеною властивiстю описали автори в статтi [52].…”