Extensions, matched products, and simple braces

By D. Bachiller. Journal of Pure and Applied Algebra, 2015

65 citations recorded by the source corpus.

We show how to construct all the extensions of left braces by ideals with trivial structure. This is useful to find new examples of left braces. But, to do so, we must know the basic blocks for extensions: the left braces with no ideals except the trivial and the total ideal, called simple left braces. In this article, we present the first non-trivial examples of finite simple left braces. To explicitly construct them, we define the matched product of two left braces, which is a useful method to recover a finite left brace from its Sylow subgroups.

Subfields: Algebra and Number Theory, Organic Chemistry

View Paper: https://www.semanticscholar.org/paper/108a066a284017417edd625cbc2be214dff2cfa3

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In Algebra and Number Theory

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Citations in the corpus

Skew left braces of nilpotent type (2018, background): “eger n, there exists a ﬁnite perfect left brace B such that Soc(B/Socn(B)) 6= {0}. Proof. We shall prove the result by induction on n. For n = 1, let B 0 be a ﬁnite simple non-trivial left brace (see [3]). Then by Theorem 3.2, there exists a perfect ﬁnite left brace B 1 with non-zero socle. By Theor”
Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation (2016, methodology): “n (Z/(q))⋆, and pand qare distinct prime numbers. Because of the Sylow theorems, it is easy to see that either a Sylow p-subgroup or a Sylow q-subgroup of Gis a normal subgroup. By Proposition 6.1 in [3], every normal Sylow subgroup of the multiplicative group of a left brace Bis an ideal of B. Ther”
On skew braces (with an appendix by N. Byott and L. Vendramin) (2017, background): “is the set of functions which are polynomial functions from N[x] with zero constant term. 2.5. Matched pair of skew braces. The construction of matched pair of braces was ﬁrst considered by Bachiller [2] for classical braces. Deﬁnition 2.32. A pair of skew braces (A,B) is said to be matched if there”