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On the multiplicative group of a two-sided skew brace of solvable type

Marco Damele

Abstract

We prove that if $(B,+,\cdot)$ is a two-sided skew brace whose additive group is solvable, then every finite quotient of the multiplicative group $(B,\cdot)$ is solvable. In particular, our result recovers Nasybullov's theorem in the finite case ~\cite[Theorem~4.3(1)]{Nas} and extends it to arbitrary two-sided skew braces of solvable type.

On the multiplicative group of a two-sided skew brace of solvable type

Abstract

We prove that if is a two-sided skew brace whose additive group is solvable, then every finite quotient of the multiplicative group is solvable. In particular, our result recovers Nasybullov's theorem in the finite case ~\cite[Theorem~4.3(1)]{Nas} and extends it to arbitrary two-sided skew braces of solvable type.

Paper Structure

This paper contains 1 theorem, 10 equations.

Key Result

Theorem 1

Let $(B,+,\cdot)$ be a two-sided skew brace such that $(B,+)$ is solvable. Then, for every normal subgroup $N\trianglelefteq (B,\cdot)$ of finite index, the quotient $(B,\cdot)/N$ is solvable.

Theorems & Definitions (2)

  • Theorem 1
  • proof