A generalization of Ito's theorem to skew braces
| Autor: | Tsang, Cindy |
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| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | J. Algebra 642 (2024), 367-399 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.jalgebra.2023.12.012 |
| Popis: | The famous theorem of It\^{o} in group theory states that if a group $G=HK$ is the product of two abelian subgroups $H$ and $K$, then $G$ is metabelian. We shall generalize this to the setting of a skew brace $(A,{\cdot\,},\circ)$. Our main result says that if $A = BC$ or $A = B\circ C$ is the product of two trivial sub-skew braces $B$ and $C$ which are both left and right ideals in the opposite skew brace of $A$, then $A$ is meta-trivial. One can recover It\^{o}'s Theorem by taking $A$ to be an almost trivial skew brace. Comment: 35 pages; Proposition 2.4 and Corollary 2.6 were modified (there was a small mistake in the proof in version 1), and the hypothesis in Lemma 6.4 was slightly relaxed. Also fixed some typos |
| Databáze: | arXiv |
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