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A generalization of Ito's theorem to skew braces

Cindy Tsang

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.

A generalization of Ito's theorem to skew braces

Abstract

The famous theorem of Itô in group theory states that if a group is the product of two abelian subgroups and , then is metabelian. We shall generalize this to the setting of a skew brace . Our main result says that if or is the product of two trivial sub-skew braces and which are both left and right ideals in the opposite skew brace of , then is meta-trivial. One can recover Itô's Theorem by taking to be an almost trivial skew brace.
Paper Structure (11 sections, 31 theorems, 184 equations)

This paper contains 11 sections, 31 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Basic facts about skew braces
  3. Products of trivial skew braces
  4. Proof of Theorem \ref{['thm:left']}
  5. Proof of Theorem \ref{['thm:right']}
  6. Some preliminary calculations
  7. Proof of Theorem \ref{['thm:ito']}
  8. A construction of ditrivial skew braces
  9. Two families of ditrival skew braces
  10. First family
  11. Second family

Key Result

Proposition 1.1

Let $G=HK$ be a group which is a product of two abelian normal subgroups $H$ and $K$. Then $[[G,G],G] =1$, namely $G$ is nilpotent of class at most two.

Theorems & Definitions (62)

  • Proposition 1.1
  • proof
  • Proposition 1.2
  • proof
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • ...and 52 more