Finite skew braces satisfy Schur-Zassenhaus for Hall ideals with complements and Sylow's third theorem on the count of Sylow p-sub-skew braces, with counterexamples for arbitrary sub-skew braces.
Analogues of Sylow's first theorem, Cauchy's theorem, and Hall's theorem for skew braces
4 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We establish an unconditional analogue of Sylow's first theorem for finite skew braces, and deduce an analogue of Cauchy's theorem. We also prove an analogue of the existence part of Hall's theorem for finite skew braces with soluble additive and multiplicative groups. We make some observations regarding the number of Sylow subskew braces of a skew brace in various cases. By applying these results we streamline the classification of skew braces of order $ pq $, where $ p,q $ are distinct prime numbers.
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math.GR 4years
2026 4representative citing papers
Every ideal of coprime order in a finite skew brace admits a complement that is a sub-skew brace.
For finite left nilpotent skew braces, multiplicative Sylow p-subgroups are always Sylow p-subbraces, removing the solvability assumption from prior work.
Sylow and Hall theorems for finite skew braces are direct consequences of the Sylow and Hall structures of finite trifactorised groups.
citing papers explorer
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The Schur--Zassenhaus Theorem and Sylow's Third Theorem for Finite Skew Braces
Finite skew braces satisfy Schur-Zassenhaus for Hall ideals with complements and Sylow's third theorem on the count of Sylow p-sub-skew braces, with counterexamples for arbitrary sub-skew braces.
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A Schur--Zassenhaus Theorem for Finite Skew Braces
Every ideal of coprime order in a finite skew brace admits a complement that is a sub-skew brace.
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Sylow theory and the nilpotency class of left nilpotent skew braces
For finite left nilpotent skew braces, multiplicative Sylow p-subgroups are always Sylow p-subbraces, removing the solvability assumption from prior work.
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On finite trifactorised groups and Sylow and Hall theorems for skew braces
Sylow and Hall theorems for finite skew braces are direct consequences of the Sylow and Hall structures of finite trifactorised groups.