Skew braces of size pq

Abstract

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

Skew braces of squarefree order

“…Remark 7.4. This case was recently considered by Acri and Bonatto [AB20], and our results agree with theirs.…”

Extensions and Well's type exact sequence of skew braces

“…It is easy to check that ( 14) and (15) gives the associativity of '+' and '•', respectively, which is enough to see that (H × I, +) and (H × I, •) are groups and (18) proves that '+' and '•' defined here satisfy (1). We denote this left skew brace structure by (H, I, χ, β, τ ).…”

“…Example 3 Let H = Z 8 be trivial skew brace and I = Z 3 × Z 2 be skew brace of order 6 defined in [1] by the following operations…”

Konstruksi Brace Dua Sisi Dengan Menggunakan Ring Jacobson

“…Adapun langkah-langkah dalam penelitian ini adalah dijelaskan dalam beberapa tahapan berikut ini. [14], [17] dan [18]. Selain itu, hubungan braces dengan struktur grup juga diberikan oleh [19].…”