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.
Sylow theory and the nilpotency class of left nilpotent skew braces
2 Pith papers cite this work. Polarity classification is still indexing.
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abstract
Let $X$ be a finite left nilpotent skew brace and let $p$ be a prime dividing $|X|$. We show that every Sylow $p$-subgroup of the multiplicative group $(X,\cdot)$ is a Sylow $p$-subbrace of $X$, and that every $p$-subbrace of $X$ is contained in some Sylow $p$-subbrace. This extends a recent result of Caranti, Del Corso, Di Matteo, Ferrara, and Trombetti by removing the solvability assumption. As an application, we obtain an upper bound for the left nilpotency class of $X$ in terms of the left nilpotency classes of its Sylow $p$-subbraces.
fields
math.GR 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Proves that in a finite skew brace B, any ideal I with |I| coprime to |B/I| admits a complement in B.
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.