Finite groups with soluble centralisers

In this talk, we present results on finite groups in which the centralisers of non-central π-elements are soluble, for a given set of primes π. Using a range of structural and reduction techniques, we determine the possible non-abelian sections of such groups. In particular, we characterize the case where all non-central elements of the group have soluble centralisers and the case π = {2}. These results also yield applications to solubility criteria arising from the non-commuting graph of a finite group. 

This is a joint work with Carmine Monetta and Gareth Tracey, started during a "Research in pairs" visit at CIRM in Trento.