arXiv subjects

Cristina Acciarri

Publications and source records attributed to Cristina Acciarri.

At least 19 recordsLinked to original sources

arXiv2026

Rank type conditions on commutators in finite groups

For a subgroup $S$ of a group $G$, let $I_G(S)$ denote the set of commutators $[g,s]=g^{-1}g^s$, where $g\in G$ and $s\in S$, so that $[G,S]$ is the subgroup generated by $I_G(S)$. We prove that if $G$ is a $p$-soluble finite group with a Sylow $p$-subgroup $P$ such that any subgroup generated by a subset of $I_G(P)$ is $r$-generated, then $[G,P]$ has $r$-bounded rank. We produce examples showing that such a result does not hold without the assumption of $p$-solubility. Instead, we prove that if a finite group $G$ has a Sylow $p$-subgroup $P$ such that (a) any subgroup generated by a subset of $I_G(P)$ is $r$-generated, and (b) for any $x\in I_G(P)$, any subgroup generated by a subset of $I_G(x)$ is $r$-generated, then $[G,P]$ has $r$-bounded rank. We also prove that if $G$ is a finite group such that for every prime $p$ dividing $|G|$ for any Sylow $p$-subgroup $P$, any subgroup generated by a subset of $I_G(P)$ can be generated by $r$ elements, then the derived subgroup $G'$ has $r$-bounded rank. As an important tool in the proofs, we prove the following result, which is also of independent interest: if a finite group $G$ admits a group of coprime automorphisms $A$ such that any subgroup generated by a subset of $I_G(A)$ is $r$-generated, then the rank of $[G,A]$ is $r$-bounded.

math.GR↗

arXiv2026

Local--global generation property of commutators in finite $π$-soluble groups

For a group $A$ acting by automorphisms on a group $G$, let $I_G(A)$ denote the set of commutators $[g,a]=g^{-1}g^a$, where $g\in G$ and $a\in A$, so that $[G,A]$ is the subgroup generated by $I_G(A)$. We prove that if $A$ is a $π$-group of automorphisms of a $π$-soluble finite group $G$ such that any subset of $I_G(A)$ generates a subgroup that can be generated by $r$ elements, then the rank of $[G,A]$ is bounded in terms of $r$. Examples show that such a result does not hold without the assumption of $π$-solubility. Earlier we obtained this type of results for groups of coprime automorphisms and for Sylow $p$-subgroups of $p$-soluble groups.

math.GR↗

arXiv2026

Profinite groups with restricted centralizers of powers

A group $G$ is said to have restricted centralizers if for every $x\in G$ the centralizer $C_G(x)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Here we take interest in profinite groups $G$ for which there is an integer $n$ such that $C_G(x^n)$ is either finite or open whenever $x\in G$. It is shown that such a group $G$ has an open normal subgroup $T$ with the property that $G/Z(T)$ has finite exponent.

math.GR↗

arXiv2026

Coprime commutators in profinite groups

By a coprime commutator in a profinite group $G$ we mean any element of the form $[x, y]$, where $x,y\in G$ and $(|x|,|y|)=1$. It is well-known that the subgroup generated by the coprime commutators of $G$ is precisely the pronilpotent residual $γ_\infty(G)$. There are several recent works showing that finiteness conditions on the set of coprime commutators have strong impact on the properties of $γ_\infty(G)$ and, more generally, on the structure of $G$. In this paper we show that if the set of coprime commutators of a profinite group $G$ is covered by countably many procyclic subgroups, then $γ_\infty(G)$ is finite-by-procyclic. In particular, it follows that $G$ is finite-by-pronilpotent-by-abelian.

math.GR↗

arXiv2024

Varieties of groups and the problem on conciseness of words

A group-word $w$ is concise in a class of groups $\mathcal X$ if and only if the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values in a group $G\in \mathcal X$. It is a long-standing open problem whether every word is concise in residually finite groups. In this paper we observe that the conciseness of a word $w$ in residually finite groups is equivalent to that in the class of virtually pro-$p$ groups. This is used to show that if $q,n$ are positive integers and $w$ is a multilinear commutator word, then the words $w^q$ and $[w^q,_{n} y]$ are concise in residually finite groups. Earlier this was known only in the case where $q$ is a prime power. In the course of the proof we establish that certain classes of groups satisfying the law $w^q\equiv1$, or $[δ_k^q,{}_n\, y]\equiv1$, are varieties.

math.GR↗

arXiv2022

Criteria for solubility and nilpotency of finite groups with automorphisms

Let $G$ be a finite group admitting a coprime automorphism $α$. Let $J_G(α)$ denote the set of all commutators $[x,α]$, where $x$ belongs to an $α$-invariant Sylow subgroup of $G$. We show that $[G,α]$ is soluble or nilpotent if and only if any subgroup generated by a pair of elements of coprime orders from the set $J_G(α)$ is soluble or nilpotent, respectively.

math.GR↗

arXiv2022

Coprime automorphisms of finite groups

Let $G$ be a finite group admitting a coprime automorphism $α$ of order $e$. Denote by $I_G(α)$ the set of commutators $g^{-1}g^α$, where $g\in G$, and by $[G,α]$ the subgroup generated by $I_G(α)$. We study the impact of $I_G(α)$ on the structure of $[G,α]$. Suppose that each subgroup generated by a subset of $I_G(α)$ can be generated by at most $r$ elements. We show that the rank of $[G,α]$ is $(e,r)$-bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of $I_G(α)$ has odd order, then $[G,α]$ has odd order too. Further, if every pair of elements from $I_G(α)$ generates a soluble, or nilpotent, subgroup, then $[G,α]$ is soluble, or respectively nilpotent.

math.GR↗

arXiv2021

Profinite groups with restricted centralizers of $π$-elements

A group $G$ is said to have restricted centralizers if for each $g$ in $G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Given a set of primes $π$, we take interest in profinite groups with restricted centralizers of $π$-elements. It is shown that such a profinite group has an open subgroup of the form $P\times Q$, where $P$ is an abelian pro-$π$ subgroup and $Q$ is a pro-$π'$ subgroup. This significantly strengthens a result from our earlier paper.

math.GR↗

arXiv2021

A stronger form of Neumann's BFC-theorem

Given a group $G$, we write $x^G$ for the conjugacy class of $G$ containing the element $x$. A famous theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the derived group $G'$ is finite. We establish the following result. Let $n$ be a positive integer and $K$ a subgroup of a group $G$ such that $|x^G|\leq n$ for each $x\in K$. Let $H=\langle K^G\rangle$ be the normal closure of $K$. Then the order of the derived group $H'$ is finite and $n$-bounded. Some corollaries of this result are also discussed.

math.GR↗

arXiv2020

Graphs encoding the generating properties of a finite group

Assume that $G$ is a finite group. For every $a, b \in\mathbb N,$ we define a graph $Γ_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and $(y_1,\dots,y_b)$ are adjacent if and only if $\langle x_1,\dots,x_a,y_1,\dots,y_b \rangle =G.$ We study several properties of these graphs (isolated vertices, loops, connectivity, diameter of the connected components) and we investigate the relations between their properties and the group structure, with the aim of understanding which information about $G$ are encoded by these graphs.

math.GR↗

arXiv2020

Genus, thickness and crossing number of graphs encoding the generating properties of finite groups

Assume that $G$ is a finite group and let $a$ and $b$ be non-negative integers. We define an undirected graph $Γ_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and $(y_1,\dots,y_b)$ are adjacent if and only $\langle x_1,\dots,x_a,y_1,\dots,y_b \rangle =G.$ Our aim is to estimate the genus, the thickness and the crossing number of the graph $Γ_{a,b}(G)$ when $a$ and $b$ are positive integers.

math.GR↗

arXiv2019

On groups in which Engel sinks are cyclic

For an element $g$ of a group $G$, an Engel sink is a subset $\mathcal{E}(g)$ such that for every $ x\in G $ all sufficiently long commutators $ [x,g,g,\ldots,g] $ belong to $\mathcal{E}(g)$. We conjecture that if $G$ is a profinite group in which every element admits a sink that is a procyclic subgroup, then $G$ is procyclic-by-(locally nilpotent). We prove the conjecture in two cases -- when $G$ is a finite group, or a soluble pro-$p$ group.

math.GR↗

arXiv2019

Engel-like conditions in fixed points of automorphisms of profinite groups

Let $q$ be a prime and $A$ an elementary abelian $q$-group acting as a coprime group of automorphisms on a profinite group $G$. We show that if $A$ is of order $q^2$ and some power of each element in $C_G(a)$ is Engel in $G$ for any $a\in A^{\#}$, then $G$ is locally virtually nilpotent. Assuming that $A$ is of order $q^3$ we prove that if some power of each element in $C_G(a)$ is Engel in $C_G(a)$ for any $a\in A^{\#}$, then $G$ is locally virtually nilpotent. Some analogues consequences of quantitative nature for finite groups are also obtained.

math.GR↗

arXiv2018

The generating graph of the abelian groups

For a group $G,$ let $Γ(G)$ denote the graph defined on the elements of $G$ in such a way that two distinct vertices are connected by an edge if and only if they generate $G$. Moreover let $Γ^*(G)$ be the subgraph of $Γ(G)$ that is induced by all the vertices of $Γ(G)$ that are not isolated. We prove that if $G$ is a 2-generated non-cyclic abelian group then $Γ^*(G)$ is connected. Moreover $\mathrm{diam}(Γ^*(G))=2$ if the torsion subgroup of $G$ is non-trivial and $\mathrm{diam}(Γ^*(G))=\infty$ otherwise. If $F$ is the free group of rank 2, then $Γ^*(F)$ is connected and we deduce from $\mathrm{diam}(Γ^*(\mathbb{Z}\times \mathbb{Z}))=\infty$ that $\mathrm{diam}(Γ^*(F))=\infty.$

math.GR↗

arXiv2018

Engel sinks of fixed points in finite groups

For an element $g$ of a group $G$, an Engel sink is a subset $\mathscr{E}(g)$ such that for every $ x\in G $ all sufficiently long commutators $ [x,g,g,\ldots,g] $ belong to $\mathscr{E}(g)$. Let $q$ be a prime, let $m$ be a positive integer and $A$ an elementary abelian group of order $q^2$ acting coprimely on a finite group $G$. We show that if for each nontrivial element $a$ in $ A$ and every element $g\in C_{G}(a)$ the cardinality of the smallest Engel sink $\mathscr{E}(g)$ is at most $m$, then the order of $γ_\infty(G)$ is bounded in terms of $m$ only. Moreover we prove that if for each $a\in A\setminus \{1\}$ and every element $g\in C_{G}(a)$, the smallest Engel sink $\mathscr{E}(g)$ generates a subgroup of rank at most $m$, then the rank of $γ_\infty(G)$ is bounded in terms of $m$ and $q$ only.

math.GR↗

arXiv2017

Profinite groups and centralizers of coprime automorphisms whose elements are Engel

Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^r$ with $r\geq2$ acting on a finite $q'$-group $G$. The following results are proved. We show that if all elements in $γ_{r-1}(C_G(a))$ are $n$-Engel in $G$ for any $a\in A^\#$, then $γ_{r-1}(G)$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$, and if, for some integer $d$ such that $2^d\leq r-1$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $G$ for any $a\in A^\#$, then the $d$th derived group $G^{(d)}$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$. Assuming $r\geq 3$ we prove that if all elements in $γ_{r-2}(C_G(a))$ are $n$-Engel in $C_G(a)$ for any $a\in A^\#$, then $γ_{r-2}(G)$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$, and if, for some integer $d$ such that $2^d\leq r-2$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $C_G(a)$ for any $a\in A^\#,$ then the $d$th derived group $G^{(d)}$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$. Analogue (non-quantitative) results for profinite groups are also obtained.

math.GR↗

arXiv2017

On verbal subgroups in finite and profinite groups

Let $w$ be a multilinear commutator word. In the present paper we describe recent results that show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely (or in some cases countably) many subgroups with a prescribed property, then the verbal subgroup $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank.

math.GR↗

arXiv2017

Profinite groups and the fixed points of coprime automorphisms

The main result of the paper is the following theorem. Let $q$ be a prime and $A$ an elementary abelian group of order $q^3$. Suppose that $A$ acts coprimely on a profinite group $G$ and assume that $C_G(a)$ is locally nilpotent for each $a\in A^{\#}$. Then the group $G$ is locally nilpotent.

math.GR↗