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.
On the Sylow Theorem for Skew Braces
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abstract
We discuss the (first) Sylow theorem for certain classes of finite skew braces, proving it to hold true when the skew brace is two-sided, bi-skew, right nilpotent, $\lambda$-homomorphic or supersoluble. We also show it to hold true for soluble skew braces that are left-nilpotent, and address a number of more specialized settings, proving general Hall-type theorems.
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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.
Analogues of Sylow's first, Cauchy's, and Hall's theorems are established for finite skew braces, with application to classification of order pq examples.
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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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Analogues of Sylow's first theorem, Cauchy's theorem, and Hall's theorem for skew braces
Analogues of Sylow's first, Cauchy's, and Hall's theorems are established for finite skew braces, with application to classification of order pq examples.