Narrow Results

We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for Lie algebras. Some open problems are discussed as well.

We prove that the base size of a transitive group G with solvable point stabilizer and with trivial solvable radical is not greater than k provided the same statement holds for the group of G-induced automorphisms of each nonabelian composition factor of G.

Finite groups with an automorphism mapping a sufficiently large proportion of elements to their inverses, squares and cubes have been studied for a long time, and the gist of the results on them is that they are “close to being abelian”. In this paper, we consider finite groups $G$ such that, for a fixed but arbitrary $\rho \in (0,1]$, some automorphism of $G$ maps at least $\rho \left|G\right|$ many elements of $G$ to their inverses, squares and cubes. We will relate these conditions to some parameters that measure, intuitively speaking, how far the group $G$ is from being solvable, nilpotent or abelian; most prominently the commuting probability of $G$, i.e. the probability that two independently uniformly randomly chosen elements of $G$ commute. To this end, we will use various counting arguments, the classification of the finite simple groups and some of its consequences, as well as a classical result from character theory.