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Alex Carrazedo Dantas

Publications and source records attributed to Alex Carrazedo Dantas.

7 recordsLinked to original sources
arXiv2026

On the Self-Similarity of Permutational Wreath Products and Their Embedding into Finitely Presented Simple Groups

In this work, we study the self-similarity of permutational wreath products of the form \(A \wr_X G\), where \(A\) is a finitely generated abelian group and \(G\) is a self-similar group (the permutational wreath product \(A \wr_X G\) is also known as a lamplighter group). In the case where \(G\) is a non-torsion contracting group, we prove that, under certain conditions, the Scott--Röver--Nekrashevych group \(V_m(\mathbb{Z}^d \wr_X G)\) is finitely presented and virtually simple. Moreover, we prove that \(\mathbb{Z}^d \wr_X G\) embeds into a finitely presented simple group. Furthermore, this provides a new family of groups that satisfy the Boone--Higman conjecture.

math.GR↗
arXiv2026

Some families of locally graded groups with finitely many orbits under automorphisms

In this work, we study three families of locally graded groups with finitely many orbits under automorphisms. We prove that: (i) a residually finite group with finitely many orbits under automorphisms is locally finite and has finite exponent; (ii) a finitely generated locally graded group with finitely many orbits under automorphisms is finite; and (iii) the Mal'cev $\mathbb{Q}$-completion of an $r$-generated free nilpotent group of class $c$ has finitely many orbits under automorphisms if and only if either $r = 2$ and $c = 3$, or $c \leq 2$

math.GR↗
arXiv2019

Soluble Groups with few orbits under automorphisms

Let $G$ be a group. The orbits of the natural action of Aut$(G)$ on $G$ are called ``automorphism orbits'' of $G$, and the number of automorphism orbits of $G$ is denoted by $ω(G)$. We prove that if $G$ is a soluble group with finite rank such that $ω(G)< \infty$, then $G$ contains a torsion-free characteristic nilpotent subgroup $K$ such that $G = K \rtimes H$, where $H$ is a finite group. Moreover, we classify the mixed order soluble groups of finite rank such that $ω(G)=3$.

math.GR↗
arXiv2019

Exponent of Self-similar finite $p$-groups

Let $p$ be a prime and $G$ a pro-$p$ group of finite rank that admits a faithful, self-similar action on the $p$-ary rooted tree. We prove that if the set $\{g\in G \ | \ g^{p^n}=1\}$ is a nontrivial subgroup for some $n$, then $G$ is a finite $p$-group with exponent at most $p^n$. This applies in particular to power abelian $p$-groups.

math.GR↗
arXiv2017

Finite Groups with 6 or 7 Automorphism Orbits

Let $G$ be a group. The orbits of the natural action of $\mbox{Aut}(G)$ on $G$ are called "automorphism orbits" of $G$, and the number of automorphism orbits of $G$ is denoted by $ω(G)$. In this paper the finite nonsolvable groups $G$ with $ω(G) \leq 6$ are classified - this solves a problem posed by Markus Stroppel - and it is proved that there are infinitely many finite nonsolvable groups $G$ with $ω(G)=7$. Moreover it is proved that for a given number $n$ there are only finitely many finite groups $G$ without nontrivial abelian normal subgroups and such that $ω(G) \leq n$, generalizing a result of Kohl.

math.GR↗
arXiv2015

On finite groups with few automorphism orbits

Denote by $ω(G)$ the number of orbits of the action of $Aut(G)$ on the finite group $G$. We prove that if $G$ is a finite nonsolvable group in which $ω(G) \leqslant 5$, then $G$ is isomorphic to one of the groups $A_5,A_6,PSL(2,7)$ or $PSL(2,8)$. We also consider the case when $ω(G) = 6$ and show that if $G$ is a nonsolvable finite group with $ω(G) = 6$, then either $G \simeq PSL(3,4)$ or there exists a characteristic elementary abelian $2$-subgroup $N$ of $G$ such that $G/N \simeq A_5$.

math.GR↗