We introduce various probabilistic finiteness conditions for profinite groups related to positive finite generation (PFG). investigate completed group rings which are PFG as modules, and use this answer [17, Question 1.2] on positively finitely groups. Using the theory of projective covers, we define characterise a version FPn property groups, called PFPn. Finally, prove how these previously de...

The theory of special groups is an axiomatization of the algebraic theory of quadratic forms, introduced by Dickmann and Miraglia (see [4]). The class of special groups, together with its morphisms, forms a category. As for other such axiomatisations, the main examples of special groups are provided by fields, in this case by applying the special group functor, which associates to each field F ...

We introduce the notion of a pro-fusion system on a pro-p group, which generalizes the notion of a fusion system on a finite p-group. We also prove a version of Alperin’s Fusion Theorem for pro-fusion systems.