arXiv subjects

Mattia Brescia

Publications and source records attributed to Mattia Brescia.

5 recordsLinked to original sources

arXiv2026

A torsion-free group of nilpotency class six with all subgroups subnormal of defect at most $5$

Casolo asked whether a torsion-free group in which every subgroup is subnormal of defect at most $n$ must be nilpotent of class at most $n$. The answer is known to be positive for $n\leq 4$. We construct a $2$-generated torsion-free nilpotent group $G$ such that $$ \cl(G)=6 \qquad\text{and}\qquad [G,{}_5H]\leq H \quad\text{for every }H\leq G. $$ Thus Casolo's question has a negative answer for $n=5$.

math.GR↗

arXiv2026

An antichain condition for infinite groups

Let $χ$ be a subgroup-theoretical property. We introduce an \emph{antichain condition} $\operatorname{ac}_χ$ which forbids the existence of infinite antichains of mutually permutable non-$χ$ subgroups whose infinite joins remain non-$χ$. This is a ''width'' analogue of the real chain condition on non-$χ$ subgroups, and it extends the usual hierarchy of weak chain conditions (double chain condition, deviation, and $\operatorname{RCC}$). Our main results show that, within the universe of generalized radical groups, the antichain condition is as rigid as the corresponding chain conditions. For the properties $χ$ of normality, almost normality, near normality, permutability, modularity, and pronormality, we prove that a generalized radical group satisfies $\operatorname{ac}_χ$ if and only if it satisfies $\operatorname{RCC}$ on non-$χ$ subgroups; equivalently, it satisfies any of the standard weak chain conditions on non-$χ$ subgroups. In particular, we obtain minimax-type dichotomies: either the group is minimax, or \emph{every} subgroup satisfies $χ$. This yields characterizations in terms of Dedekind groups, quasi-Hamiltonian groups, groups with modular subgroup lattice, and $\overline{T}$-groups. In the pronormal case, one has to deal with locally finite simple groups and a use of the Classification of Finite Simple Groups seems unavoidable.

math.GR↗

arXiv2026

A Torsion-free Supersoluble Group with Trivial Outer Automorphism Group

We give a negative solution to Problem~13.23 of the Kourovka Notebook. We construct a torsion-free group $G$ of Hirsch length $14$ admitting a finite series \[ 1=G_0\triangleleft G_1\triangleleft\cdots\triangleleft G_{14}=G \] in which every $G_i$ is normal in $G$ and every factor is infinite cyclic, but such that $\Out(G)=1$.

math.GR↗

arXiv2025

The Pseudocentre of a Group (with an appendix by Anthony Genevois)

In 1973, Jim Wiegold introduced the concept of pseudocentre P(G) of a group G as the intersection of the normal closures of the centralizers of its elements. He proved that the pseudocentre of a non-trivial finite group is always non-trivial, giving a new variable on which one can use induction in finite group theory. In the same paper, Wiegold states that no obvious relations seem to hold between the pseudocentre and the canonical characteristic subgroups of a group. The aim of this work is to show that the pseudocentre is indeed much more involved in the structure of an arbitrary group then anyone could have expected. For example, we prove that a soluble group coincides with its pseudocentre if and only if it is abelian, and that the structure of the commutator subgroup strongly influences the structure of the pseudocentre. And this is not the end of the story. In fact, the behaviour of the pseudocentre in arbitrary (possibly infinite) groups can be extremely wild: sometimes it is very difficult even to understand whether the pseudocentre is trivial or not. This wilderness is exampled by some of our main results (see the introduction for a complete list): 1) There exists a polycyclic group of Hirsch length 3 in which the pseudocentre is trivial. 2) The pseudocentre of the group of unitriangular matrices over any field is the largest term of the upper central series that is abelian. 3) Free products have a trivial pseudocentre, but there exist amalgamated free products of non-trivial groups coinciding with their pseudocentre. 4) Weakly regular branch groups have a trivial pseudocentre. 5) The pseudocentre of the Thompson group is the derived subgroup. 6) Wreath products can have a totally arbitrary pseudocentre.

math.GR↗

arXiv2024

A determinant for automorphisms of groups

Let $H$ and $K$ be groups. In this paper we introduce a concept of determinant for automorphisms of $H\times K$ and some concepts of incompatibility for group pairs as a measure of how much $H$ and $K$ are fare from being isomorphic. With the aid of the tools developed from these definitions, we give a characterisation of invertible automorphisms of $H\times K$ by means of their determinants and an explicit description of Aut($H\times K$) as a group of $2$-by-$2$ matrices, in case $H$ or $K$ belong to some relevant classes of groups. Many theoretical and practical applications of the determinants will be presented, together with examples and an analysis on some computational advantages of the determinants.

math.GR↗