Skew left braces of nilpotent type

Cedó, Ferran; Smoktunowicz, Agata; Vendramin, L.

doi:10.1112/plms.12209

Cited by 67 publications

(113 citation statements)

References 40 publications

(82 reference statements)

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“…where 1 ≤ a ≤ p − 1 and in particular G is abelian. According to [9,Corollary 4.3], since both the additive and the multiplicative group of the skew brace associated to G are abelian, then such skew brace splits as a direct product of the skew brace of size p 2 and a trivial skew brace of size q.…”

Section: Regular Subgroupsmentioning

confidence: 99%

Skew braces of size pq

Acri

Bonatto

2020

Communications in Algebra

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In this paper we enumerate the skew braces of size p 2 q for p, q odd primes by the classification of regular subgroups of the holomorph of the groups of size p 2 q. In particular, we provide explicit formulas for the skew braces of abelian type.Proof. The groups H, L and G c,θ are regular. If G c,β 0,r and G d,β 0,r are conjugate by some h ∈ Aut(A), then h normalizes β 0,r and so it centralizes it. So h(σ) c β 0,r = σ c β 0,r (mod ǫ, τ ). Then σ c β 0,r = σ d β 0,r (mod ǫ, τ ) which implies that c = d. The same argument applies if θ = α 1,1 β 0,r .Let G be a regular subgroup of Hol(A) such that |π 2 (G)| = p. According to Table 16 we need to discuss three cases.Assume that π 2 (G) = α 1,1 . Then the kernel has order pq and so, up to conjugation by a power of α 1,1 we can assume that G = ǫ, σ n τ m , σ a τ b α 1,1 . By condition (K) we have n = 0. So G = ǫ, τ, σ a α 1,1 and G is conjugate to H by α 0,a −1 .Assume that π 2 (G) = β 0,r . Then G has the following standard presentation:If n = 0, we conjugate G by α − m n ,1 and then by α 0,b −1 and we get L. If n = 0 we have G = ǫ, τ, σ b β 0,r = G b,β 0,r .Assume that π 2 (G) = α 1,1 β 0,r . Then

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Section: Regular Subgroupsmentioning

confidence: 99%

Skew braces of size pq

Acri

Bonatto

2020

Communications in Algebra

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“…Since A is a non-trivial skew left brace, 0 < |A/Soc(A)| < |A|, and therefore A/Soc(A) is right nilpotent by the minimality of |A|. By [15,Proposition 2.17], A is right nilpotent, a contradiction.…”

Section: Right P-nilpotent Skew Left Bracesmentioning

confidence: 98%

“…Then a * b = (x • y) * b = x * (y * b) + y * b + x * b = x * b ∈ A p * L n (A p , A p ) since A p ′ * A p = 0. The skew left brace A p is left nilpotent by [15,Proposition 4.4], so there exists n ∈ N such that L n (A p , A p ) = 0.…”

Section: Left P-nilpotent Skew Left Bracesmentioning

confidence: 99%

See 2 more Smart Citations

Retractability of solutions to the Yang–Baxter equation and p-nilpotency of skew braces

Acri

Lutowski

Vendramin

2019

Int. J. Algebra Comput.

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Using Bieberbach groups we study multipermutation involutive solutions to the Yang-Baxter equation. We use a linear representation of the structure group of an involutive solution to study the unique product property in such groups. An algorithm to find subgroups of a Bieberbach group isomorphic to the Promislow subgroup is introduced and then used in the case of structure group of involutive solutions. To extend the results related to retractability to non-involutive solutions, following the ideas of Meng, Ballester-Bolinches and Romero, we develop the theory of right p-nilpotent skew braces. The theory of left p-nilpotent skew braces is also developed and used to give a short proof of a theorem of Smoktunowicz in the context of skew braces.

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Skew Braces: A Brief Survey

Vendramin

2024

Trends in Mathematics

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Skew left braces of nilpotent type

Cited by 67 publications

References 40 publications

Skew braces of size pq

Skew braces of size pq

Retractability of solutions to the Yang–Baxter equation and p-nilpotency of skew braces

Skew Braces: A Brief Survey

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