Triply factorised groups and nearrings

Hubert, Peter

doi:10.1017/cbo9780511721205.014

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Corollary 33 Let a Be A Skew Brace And R A Its Associated So...1

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2018

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“…Let ι : X → G(X, r) be the canonical map, i.e., ι(x) = e x . In general, ι is not injective: 12) and τ = (34). Then (X, r), r(x, y) = (σ(y), τ (x)), is a non-degenerate solution of the YBE.…”

Section: Corollary 33 Let a Be A Skew Brace And R A Its Associated So...mentioning

confidence: 99%

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

2018

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Braces are generalizations of radical rings, introduced by Rump to study involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation (YBE). Skew braces were also recently introduced as a tool to study not necessarily involutive solutions. Roughly speaking, skew braces provide group-theoretical and ring-theoretical methods to understand solutions of the YBE. It turns out that skew braces appear in many different contexts, such as near-rings, matched pairs of groups, triply factorized groups, bijective 1-cocycles and Hopf-Galois extensions. These connections and some of their consequences are explored in this paper. We produce several new families of solutions related in many different ways with rings, near-rings and groups. We also study the solutions of the YBE that skew braces naturally produce. We prove, for example, that the order of the canonical solution associated with a finite skew brace is even: it is two times the exponent of the additive group modulo its center.

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Section: Corollary 33 Let a Be A Skew Brace And R A Its Associated So...mentioning

confidence: 99%

“…Let X = {1, 2, 3, 4}, σ = (12) and τ =(34). Then (X, r), r(x, y) = (σ(y), τ (x)), is a non-degenerate solution of the YBE.…”

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confidence: 99%