Idealisation of Skew Left Braces

Idealisation is the natural brace-theoretical analogue of normalisation in groups. Previous works have shown that idealisers do not exist for sub-skew braces in general. In this paper, we prove that they do exist for left ideals. We present a generalisation of the celebrated Frattini’s argument for skew left braces of nilpotent type. We also use this to introduce a Sylow system idealiser for nilpotent-type skew left braces, a generalisation of the Sylow system normaliser, and prove generalisations of some of its properties. Finally, we prove that the fixed ideals of a nilpotent-type skew left brace are contained in the hypercentre.