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Nadia Mazza

Publications and source records attributed to Nadia Mazza.

15 recordsLinked to original sources
arXiv2026

On cohomological dimensions of totally disconnected locally compact groups

In this paper, we introduce Mackey functors for a t.d.l.c. group and define the cohomological dimension of this group over the Mackey category. We then compare this dimension to the rational discrete cohomological dimension defined by Castellano and Weigel, as well as to the Bredon cohomological dimension of that t.d.l.c. group with respect to the family of compact open subgroups. We also extend results about the geometric dimension of a t.d.l.c. group.

math.GR↗
arXiv2025

The Euler characteristic of an endotrivial complex

Let $G$ be a finite group and $k$ a field of prime characteristic $p$. We examine the Lefschetz homomorphism $\Lambda: \mathcal{E}_k(G) \to O(T(kG))$ from the group of endotrivial complexes, i.e. the Picard group of the bounded homotopy category of $p$-permutation modules $K^b({}_{kG}\mathbf{triv})$, to the orthogonal unit group of the Grothendieck group of $K^b({}_{kG}\mathbf{triv})$, i.e. the trivial source ring. When $p = 2$ and $k = \mathbb{F}_2$, $\Lambda$ is surjective when $G$ has a Sylow $2$-subgroup with fusion controlled by its normalizer, and when $G$ has dihedral Sylow $2$-subgroups. When $p$ is odd, $\Lambda$ is surjective if $G$ has a cyclic Sylow $p$-subgroup or is $p$-nilpotent, but we exhibit examples of groups of $p$-rank 2 or greater for which $\Lambda$ is not surjective. We also examine the kernel of the Lefschetz homomorphism, determining it for all groups when $p = 2$ and for groups with cyclic Sylow $p$-subgroups when $p$ is odd.

math.RT↗
arXiv2025

Burnside rings for racks and quandles

We restructure and advance the classification theory of finite racks and quandles by employing powerful methods from transformation groups and representation theory, especially Burnside rings. These rings serve as universal receptacles for those invariants of racks and quandles that are additive with respect to decompositions. We present several fundamental results regarding their structure, including additive bases and multiplicative generators. We also develop a theory of marks, which is analogous to counting fixed points of group actions and computing traces in character theory, and which is comprehensive enough to distinguish different elements in the Burnside rings. The new structures not only offer a fresh framework for the classification theory of finite racks and quandles but also equip us with tools to develop these ideas and create interfaces that strengthen connections with related areas of algebra. For example, they extend the Dress--Siebeneicher theory of the Burnside ring of the infinite cyclic group beyond the realm of permutation racks.

math.RT↗
arXiv2025

On the orbit category on nontrivial $p$-subgroups and endotrivial modules

Let $p$ be a prime, let $G$ be a finite group of order divisible by $p$, and let $k$ be a field of characteristic $p$. An endotrivial $kG$-module is a finitely generated $kG$-module $M$ such that its endomorphism algebra $\operatorname{End}_kM$ decomposes as the direct sum of a one-dimensional trivial $kG$-module and a projective $kG$-module. In this article, we determine the fundamental group of the orbit category on nontrivial $p$-subgroups of $G$ for a large class of finite groups, and use Grodal's approach to describe the group of endotrivial modules for such groups. Hence, we improve on the results about the group of endotrivial modules for finite groups with abelian Sylow $p$-subgroups obtained by Carlson and Th\'evenaz. With some additional analysis, we then determine the fundamental group of the orbit category on nontrivial $p$-subgroups of $G$ and the group of endotrivial $kG$-modules in the case when $G$ has a metacyclic Sylow $p$-subgroup for $p$ odd.

math.RT↗
arXiv2022

On the (crossed) Burnside ring of profinite groups

In this paper we investigate some properties of the Burnside ring of a profinite group as defined in \cite{ds}. We introduce the notion of the crossed Burnside ring of a profinite FC-group, and generalise some results from finite to profinite (FC-)groups. In our investigations, we also obtain results on profinite FC-groups which may be of independent interest.

math.GR↗
arXiv2021

Torsion Free Endotrivial Modules for Finite Groups of Lie Type

In this paper we determine the torsion free rank of the group of endotrivial modules for any finite group of Lie type, in both defining and non-defining characteristic. On our way to proving this, we classify the maximal rank $2$ elementary abelian $\ell$-subgroups in any finite group of Lie type, for any prime $\ell$, which may be of independent interest.

math.GR↗
arXiv2021

On the cohomology of pro-fusion systems

We prove the Cartan-Eilenberg stable elements theorem and construct a Lyndon-Hochschild-Serre type spectral sequence for pro-fusion systems. As an application, we determine the continuous mod-$p$ cohomology ring of $\text{GL}_2(\mathbb{Z}_p)$ for any odd prime $p$.

math.AT↗
arXiv2020

A geometric approach to Quillen's conjecture

We introduce {\em admissible collections} for a finite group $G$ and use them to prove that most of the finite classical groups in non-defining characteristic satisfy the {\em Quillen dimension at $p$ property}, a strong version of Quillen's conjecture, at a given odd prime divisor $p$ of $|G|$. Compared to the methods in \cite{AS1993}, our techniques are simpler.

math.GR↗
arXiv2019

The stable category and invertible modules for infinite groups

We construct a well-behaved stable category of modules for a large class of infinite groups. We then consider its Picard group, which is the group of invertible (or endotrivial) modules. We show how this group can be calculated when the group acts on a tree with finite stabilisers.

math.GR↗
arXiv2017

The pro-$p$ group of upper unitriangular matrices

We study the pro-$p$ group $G$ whose finite quotients give the prototypical Sylow $p$-subgroup of the general linear groups over a finite field of prime characteristic $p$. In this article, we extend the known results on the subgroup structure of $G$. In particular, we give an explicit embedding of the Nottingham group as a subgroup and show that it is selfnormalising. Holubowski (\cite{holub1,holub0,holub2}) studies a free product $C_p*C_p$ as a (discrete) subgroup of $G$ and we prove that its closure is selfnormalising of infinite index in the subgroup of $2$-periodic elements of $G$. We also discuss change of rings: field extensions and a variant for the $p$-adic integers, this latter linking $G$ with some well known $p$-adic analytic groups. Finally, we calculate the Hausdorff dimensions of some closed subgroups of $G$ and show that the Hausdorff spectrum of $G$ is the whole interval $[0,1]$ which is obtained by considering partition subgroups only.

math.GR↗
arXiv2015

Endotrivial Modules for Finite Groups of Lie Type A in Nondefining Characteristic

Let $G$ be a finite group such that $\text{SL}(n,q)\subseteq G \subseteq \text{GL}(n,q)$ and $Z$ be a central subgroup of $G$. In this paper we determine the group $T(G/Z)$ consisting of the equivalence classes of endotrivial $k(G/Z)$-modules where $k$ is an algebraically closed field of characteristic $p$ such that $p$ does not divide $q$. The results in this paper complete the classification of endotrivial modules for all finite groups of Lie Type $A$, initiated earlier by the authors.

math.GR↗
arXiv2015

Endotrivial modules for the sporadic simple groups and their covers

In a step towards the classification of endotrivial modules for quasi-simple groups, we investigate endotrivial modules for the sporadic simple groups and their covers. A main outcome of our study is the existence of torsion endotrivial modules with dimension greater than one for several sporadic groups with $p$-rank greater than one.

math.GR↗
arXiv2014

Endotrivial Modules for the General Linear Group in a Nondefining Characteristic

Suppose that $G$ is a finite group such that $\operatorname{SL}(n,q)\subseteq G \subseteq \operatorname{GL}(n,q)$, and that $Z$ is a central subgroup of $G$. Let $T(G/Z)$ be the abelian group of equivalence classes of endotrivial $k(G/Z)$-modules, where $k$ is an algebraically closed field of characteristic~$p$ not dividing $q$. We show that the torsion free rank of $T(G/Z)$ is at most one, and we determine $T(G/Z)$ in the case that the Sylow $p$-subgroup of $G$ is abelian and nontrivial. The proofs for the torsion subgroup of $T(G/Z)$ use the theory of Young modules for $\operatorname{GL}(n,q)$ and a new method due to Balmer for computing the kernel of restrictions in the group of endotrivial modules.

math.GR↗
arXiv2010

On a strong form of Oliver's p-group conjecture

We introduce a strong form of Oliver's p-group conjecture and derive a reformulation in terms of the modular representation theory of a quotient group. The Sylow p-subgroups of the symmetric group S_n and of the general linear group GL_n(F_q) satisfy both the strong conjecture and its reformulation.

math.GR↗