On $π$-nilpotencies of finite skew braces

Abstract

Let $π$ be a set of primes. We extend the theory of left and right $π$-nilpotency in skew braces beyond the nilpotent type setting, and introduce the study of central $π$-nilpotency for finite skew braces. We investigate the relationships between these three notions, and a characterisation of central $π$-nilpotency in terms of chief factors is also established. This leads to the introduction of the $π$-Fitting ideal of a skew brace, for which we prove a local analogue of Fitting's theorem and provide a description in terms of centralisers of chief factors. Finally, we introduce the $π$-core of a skew brace and investigate its relationship with $π$-ideals and central $π$-nilpotency.