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
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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 3years
2026 3verdicts
UNVERDICTED 3representative citing papers
Proves that in a finite skew brace B, any ideal I with |I| coprime to |B/I| admits a complement in B.
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
Proves that in a finite skew brace B, any ideal I with |I| coprime to |B/I| admits a complement in B.
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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.