A generalization of Ito's theorem to skew braces

Abstract

The famous theorem of Itô 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 Itô's Theorem by taking $A$ to be an almost trivial skew brace.

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

Cited by in corpus (1)

• On Grün's lemma for perfect skew braces