Random elements of a free profinite group generate a free subgroup

Random Normal Subgroups of Free Profinite Groups

Diamond Theorem for a Finitely Generated Free Profinite Group

We extend Haran’s Diamond Theorem to closed subgroups of a finitely generated free profinite group. This gives an affirmative answer to Problem 25.4.9 in [FrJ].

Inverse Automata and Profinite Topologies on a Free Group

Generalized Primitive Elements of a Free Group

We study endomorphisms of a free group of finite rank by means of their action on specific sets of elements. In particular, we prove that every endomorphism of the free group of rank 2 which preserves an automorphic orbit (i.e., acts “like an automorphism” on one particular orbit), is itself an automorphism. Then, we consider elements of a different nature, defined by means of homological prope...

The generalised Fitting subgroup of a profinite group

The generalised Fitting subgroup of a finite group is the group generated by all subnormal subgroups that are either nilpotent or quasisimple. The importance of this subgroup in finite group theory stems from the fact that it always contains its own centraliser, so that any finite group is an abelian extension of a group of automorphisms of its generalised Fitting subgroup. We define a class of...