We show that the category of discrete modules over an infinite profinite group has no non-zero projective objects and does not satisfy Ab4*. also prove same types results in a generalized setting using ring with linear topology.

Our aim is to transfer several foundational results from the modular representation theory of finite groups to the wider context of profinite groups. We are thus interested in profinite modules over the completed group algebra k[[G]] of a profinite group G, where k is a finite field of characteristic p. We define the concept of relative projectivity for a profinite k[[G]]-module. We prove a cha...

This article is concerned with the representation growth of profinite groups over finite fields. We investigate structure uniformly bounded exponential (UBERG). Using crown-based powers, we obtain some necessary and sufficient conditions for to have UBERG. As an application, prove that class UBERG closed under split extensions but fails be in general. On other hand, show closely related probabi...

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...