Some results on parameters of finite groups quantitatively related to abelianity

Abstract: 

 In the literature, there are various results stating that a finite group in which a certain parameter (e.g., the number of conjugacy classes or the number of prime order subgroups) is large enough must be in some sense close to being abelian. In this talk, I give an overview on some of these results and present own results of two types: one indicating that the maximum automorphism order is such a parameter and one developing further the already well-established theory on finite groups with an automorphism inverting, squaring or cubing a relatively large proportion of elements.