Construction of Finite Braces
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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 p, q-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 p, q-groups are verified. A further example illustrates what happens beyond semidirect products.
KeywordsAffine structure Brace Metacommutation Sylow basis
Mathematics Subject Classification53B05 20F16 68R05 08A05 20E22 81R50
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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