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We prove that every finite dimensional representation of a finite group over a field of characteristic p admits a finite resolution by p-permutation modules. The proof involves a reformulation in terms of derived categories.

We establish conditions under which the fundamental group of a graph of finite $p$-groups is necessarily residually $p$-finite. The technique of proof is independent of previously established results of this type, and the result is also…

We state a conjecture on the reduction modulo the defining characteristic of a unipotent representation of a finite reductive group.

Following our previous work, we develop an algorithm to compute a presentation of the fundamental group of certain partial compactifications of the complement of a complex arrangement of lines in the projective plane. It applies, in…

This paper is a summary of author's results on finite flat commutative group schemes. The properties of the generic fibre functor are discussed. A complete classification of finite local flat commutative group schemes over mixed…

For the group of endo-permutation modules of a finite \(p\)-group, there is a surjective reduction modulo \(p\) homomorphism from a complete discrete valuation ring of characteristic 0 to its residue field of characteristic \(p\). We prove…

We introduce a new obstruction to lifting smooth proper varieties in characteristic $p>0$ to characteristic $0$. It is based on Grothendieck's specialization homomorphism and the resulting discrete finiteness properties of \'etale…

We survey new results on finite groups of birational transformations of algebraic varieties.

We reformulate several basic notions of notions in finite group theory in terms of iterations of the lifting property (orthogonality) with respect to particular morphisms. Our examples include the notions being nilpotent, solvable, perfect,…

We initiate a systematic study of the perfection of affine group schemes of finite type over fields of positive characteristic. The main result intrinsically characterises and classifies the perfections of reductive groups, and obtains a…

We obtain a new bound of certain double multiplicative character sums. We use this bound together with some other previously obtained results to obtain new algorithms for finding roots of polynomials modulo a prime $p$.

New homotopy invariant finiteness conditions on modules over commutative rings are introduced, and their properties are studied systematically. A number of finiteness results for classical homological invariants like flat dimension,…

We consider deformations of bounded complexes of modules for a profinite group G over a field of positive characteristic. We prove a finiteness theorem which provides some sufficient conditions for the versal deformation of such a complex…

Let $G$ be a finite group and let $p$ be a prime. In this paper, we study the structure of finite groups with a large number of $p$-regular conjugacy classes or, equivalently, a large number of irreducible $p$-modular representations. We…

We show that the classifying space of a $p$-local compact group is approximated by a telescope of classifying spaces of $p$-local finite groups. This result has numerous implications, like a Stable Elements Theorem for $p$-local compact…

We obtain a Bogomolov type of result for the additive group scheme in characteristic $p$. Our result is equivalent with a Bogomolov theorem for Drinfeld modules defined over a finite field.

In this note, we verify that several fundamental results from the theory of representations of reductive $p$-adic groups, extend to finite central extensions of these groups.

This survey describes some recent work, by the authors and others, on the existence of algebraic fibrations of group extensions, as well as the finiteness properties of their algebraic fibers, in the realm of both abstract and pro-$p$…

In this report we summarize this work, all finite simple groups $G$ can determined uniformly using their orders $|G|$ and the set $\pi_e(G)$ of their element orders.

This note deals with the computation of the factorization number $F_2(G)$ of a finite group $G$. By using the M\"{o}bius inversion formula, explicit expressions of $F_2(G)$ are obtained for two classes of finite abelian groups, improving…