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Construction of Finite Braces

  • Wolfgang RumpEmail author
Article
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Abstract

Affine structures of a group G (=braces with adjoint group G) are characterized equationally without assuming further invertibility conditions. If G is finite, the construction of affine structures is reduced to affine structures of pq-groups. The delicate relationship between finite solvable groups and involutive Yang–Baxter groups is further clarified by showing that much of an affine structure is already inherent in the Sylow system of the group. Semidirect products of braces are modified (shifted) in two ways to handle affine structures of semidirect products of groups. As an application, two of Vendramin’s conjectures on affine structures of pq-groups are verified. A further example illustrates what happens beyond semidirect products.

Keywords

Affine structure Brace Metacommutation Sylow basis 

Mathematics Subject Classification

53B05 20F16 68R05 08A05 20E22 81R50 

Notes

Acknowledgements

The author would like to express thanks to Lendro Vendramin for providing a gap program which was used to check the result of Example 4.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute for Algebra and Number TheoryUniversity of StuttgartStuttgartGermany

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