On skew braces (with an appendix by N. Byott and L. Vendramin)

Smoktunowicz, Agata; Vendramin, L.

doi:10.4171/jca/2-1-3

Cited by 140 publications

(164 citation statements)

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“…If the skew left brace is finite and both groups are nilpotent, then it is possible to write the skew left brace as a direct product of skew left braces of prime‐power order (see Corollary ). This result is then applied to study infinite skew left braces with multiplicative group isomorphic to

double-struckZ

(see Theorems and ); this answers [, Question A.10]. Left nilpotent skew left braces are also studied.…”

Section: Introductionmentioning

confidence: 91%

“…The wreath product of left braces appears in [, Corollary 1] (see also [, Section 4]). This construction also works for skew left braces (see [, Corollary 2.39]). Theorem Let

B

be a finite perfect skew left brace.…”

Section: Perfect Skew Left Bracesmentioning

confidence: 99%

“…Thus, Corollary implies that

A / I_{k}

is a direct product of skew left braces of prime‐power size. Using results of Kohl quoted in [, Example A.7], such skew left braces are either of abelian type or of size

2^{α}

for some

α

. Let us assume that

A

is not of abelian type and let

a, b \in A

be such that

a + b - a - b \neq 0

.…”

Section: Braces With Cyclic Multiplicative Groupmentioning

confidence: 99%

“…In [, Theorem 3.1] it is proved that for a skew left brace

A

, the map

r_{A} : A \times A \to A \times A, r_{A} false(a, b false) = false(- a + a \circ b, {false(- a + a \circ b false)}^{'} \circ a \circ b false)

is a non‐degenerate set‐theoretic solution of the Yang–Baxter equation. In [, Theorem 4.5] it is proved that if

(X, r)

be a non‐degenerate solution of the Yang–Baxter equation, then there exists a unique skew left brace structure over the group

G = G (X, r) = ⟨ X : x \circ y = u \circ v whenever r false(x, y false) = false(u, v false) ⟩

such that

where

ι : X \to G (X, r)

is the canonical map. Moreover, the pair

(G (X, r), ι)

has the following universal property: if

B

is a skew left brace and

f : X \to B

is a map such that

then there exists a unique skew left brace homomorphism

ϕ : G (X, r) \to B

such that

commute.…”

Section: Introductionmentioning

confidence: 99%

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Skew left braces of nilpotent type

Cedó

Smoktunowicz

Vendramin

2018

Proc. London Math. Soc.

Self Cite

105

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double-struckZ

(see Theorems and ); this answers [, Question A.10]. Left nilpotent skew left braces are also studied.…”

Section: Introductionmentioning

confidence: 91%

“…The wreath product of left braces appears in [, Corollary 1] (see also [, Section 4]). This construction also works for skew left braces (see [, Corollary 2.39]). Theorem Let

B

be a finite perfect skew left brace.…”

Section: Perfect Skew Left Bracesmentioning

confidence: 99%

“…Thus, Corollary implies that

A / I_{k}

is a direct product of skew left braces of prime‐power size. Using results of Kohl quoted in [, Example A.7], such skew left braces are either of abelian type or of size

2^{α}

for some

α

. Let us assume that

A

is not of abelian type and let

a, b \in A

be such that

a + b - a - b \neq 0

.…”

Section: Braces With Cyclic Multiplicative Groupmentioning

confidence: 99%

“…In [, Theorem 3.1] it is proved that for a skew left brace

A

, the map

r_{A} : A \times A \to A \times A, r_{A} false(a, b false) = false(- a + a \circ b, {false(- a + a \circ b false)}^{'} \circ a \circ b false)

is a non‐degenerate set‐theoretic solution of the Yang–Baxter equation. In [, Theorem 4.5] it is proved that if

(X, r)

be a non‐degenerate solution of the Yang–Baxter equation, then there exists a unique skew left brace structure over the group

G = G (X, r) = ⟨ X : x \circ y = u \circ v whenever r false(x, y false) = false(u, v false) ⟩

such that

where

ι : X \to G (X, r)

is the canonical map. Moreover, the pair

(G (X, r), ι)

has the following universal property: if

B

is a skew left brace and

f : X \to B

is a map such that

then there exists a unique skew left brace homomorphism

ϕ : G (X, r) \to B

such that

commute.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Skew left braces of nilpotent type

Cedó

Smoktunowicz

Vendramin

2018

Proc. London Math. Soc.

Self Cite

105

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“…Skew braces are also related to Hopf-Galois extensions as explained in [21], since they are both in connection with regular subgroups of the holomorph of a group. The results of this paper have already been partially proved in [7] (the authors claim that the missing cases are the subject of a second paper in preparation), through the connection between skew braces and Hopf-Galois extensions.…”

Section: Introductionmentioning

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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Skew Braces: A Brief Survey

Vendramin

2024

Trends in Mathematics

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On skew braces (with an appendix by N. Byott and L. Vendramin)

Cited by 140 publications

References 45 publications

Skew left braces of nilpotent type

Skew left braces of nilpotent type

Skew braces of size pq

Skew Braces: A Brief Survey

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