Counterexample to a conjecture about braces

By D. Bachiller. 2015

107 citations recorded by the source corpus.

We find an example of a finite solvable group (in fact, a finite $p$-group) without any left brace structure (equiv. which is not an IYB group). Our argument is an improvement of an argument of Rump, using previous work in other areas of Burde, and of Featherstonhaugh, Caranti and Childs, which we relate to brace theory.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/63df43a774ab8e7dbb720e705e032320b2e88a60

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Problems on skew left braces (2018, methodology): “Left braces of size p2 and p3, where p is a prime number, were classified by Bachiller in [3].”
Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation (2016, background): “But not all finite solvable groups are isomorphic to the multiplicative group of a left brace [2].”
Solutions of the Yang–Baxter equation associated to skew left braces, with applications to racks (2016, methodology): “, and that many of the previous results are more naturally stated in terms of this structure. A good introduction to left braces can be found in [20]. This new approach turned out to be very fruitful [4, 3, 5, 6, 21, 18, 26, 38, 37]. In particular, in [5] a method was presented to construct, from a ”
Skew left braces of nilpotent type (2018, methodology): “One easily proves by induction that A([1]) ⊇ A([2]) ⊇ · · · .”
Bi-skew braces and Hopf Galois structures (2019, unknown): “Generalizing [13] for (B,+) a radical ring, Bachiller [3] proved that if m+2 ≤ p, then for every b in B, the order of b in (B, ◦) is equal to the order of b in (B,+).”