Nuova ricerca
 Cristina ACCIARRI Ricercatore t.d. art. 24 c. 3 lett. BDipartimento di Scienze Fisiche, Informatiche e Matematiche sede ex-Matematica

## Pubblicazioni

2023 - Criteria for solubility and nilpotency of finite groups with automorphisms [Articolo su rivista]
Acciarri, Cristina; Guralnick, Robert M.; Shumyatsky, Pavel
abstract

2022 - COPRIME AUTOMORPHISMS OF FINITE GROUPS [Articolo su rivista]
Acciarri, C.; Guralnick, R. M.; Shumyatsky, P.
abstract

Let G be a finite group admitting a coprime automorphism α of order e. Denote by IG(α) the set of commutators g−1gα, where g ∈ G, and by [G, α] the subgroup generated by IG(α). We study the impact of IG(α) on the structure of [G, α]. Suppose that each subgroup generated by a subset of IG(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 IG(α) has odd order, then [G, α] has odd order too. Further, if every pair of elements from IG(α) generates a soluble, or nilpotent, subgroup, then [G, α] is soluble, or respectively nilpotent.

2022 - Profinite groups with restricted centralizers of $$\pi$$-elements [Articolo su rivista]
Acciarri, Cristina; Shumyatsky, Pavel
abstract

A group G is said to have restricted centralizers if for each g in G the centralizer CG(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 × 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.

2021 - A stronger form of Neumann's BFC-theorem [Articolo su rivista]
Acciarri, C; Shumyatsky, P
abstract

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| ≤ n for each x ∈ K. Let H=⟨K^G⟩ 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.

2021 - Genus, thickness and crossing number of graphs encoding the generating properties of finite groups [Articolo su rivista]
Acciarri, C.; Lucchini, A.
abstract

Assume that G is a finite group and let a and b be non-negative integers. We define an undirected graph Γa,b(G) whose vertices correspond to the elements of Ga∪Gb and in which two tuples (x1,…,xa) and (y1,…,yb) are adjacent if and only if 〈x1,…,xa,y1,…,yb〉=G. Our aim is to estimate the genus, the thickness and the crossing number of the graph Γa,b(G) when a and b are positive integers, giving explicit lower bounds on these invariants in terms of |G|.

2021 - On the rank of a finite group of odd order with an involutory automorphism [Articolo su rivista]
Acciarri, C; Shumyatsky, P
abstract

Let G be a finite group of odd order admitting an involutory automorphism ϕ, and let G-ϕ be the set of elements of G transformed by ϕ into their inverses. Note that [ G, ϕ] is precisely the subgroup generated by G-ϕ. Suppose that each subgroup generated by a subset of G-ϕ can be generated by at most r elements. We show that the rank of [ G, ϕ] is r-bounded.

2020 - Engel-like conditions in fixed points of automorphisms of profinite groups [Articolo su rivista]
Acciarri, C; Silveira, D
abstract

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 ∈ 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 ∈ A^#, then G is locally virtually nilpotent. Some analogues of quantitative nature for finite groups are also obtained.

2020 - Graphs encoding the generating properties of a finite group [Articolo su rivista]
Acciarri, C.; Lucchini, A.
abstract

Assume that G is a finite group. For every (Formula presented.), we define a graph (Formula presented.) whose vertices correspond to the elements of (Formula presented.) and in which two tuples (Formula presented.) and (Formula presented.) are adjacent if and only if (Formula presented.). 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 is encoded by these&nbsp;graphs.

2019 - Engel sinks of fixed points in finite groups [Articolo su rivista]
Acciarri, C; Shumyatsky, P; Silveira, D. S.
abstract

For an element g of a group G, an Engel sink is a subset E (g) such that for every x ∈ G all sufficiently long commutators [x,g,g,...,g] belong to 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 ∈ C_G(a) the cardinality of the smallest Engel sink E (g) is at most m, then the order of γ∞(G) is bounded in terms of m only. Moreover we prove that if for each a ∈ A {1} and every element g ∈ CG(a), the smallest Engel sink E(g) generates a subgroup of rank at most m, then the rank of γ∞(G) is bounded in terms of m and q only.

2019 - On groups in which Engel sinks are cyclic [Articolo su rivista]
Acciarri, C; Shumyatsky, P
abstract

For an element g of a group G, an Engel sink is a subset E(g) such that for every x ∈ G all sufficiently long commutators [x,g,g,...,g] belong to 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.

2019 - Profinite groups with an automorphism whose fixed points are right Engel [Articolo su rivista]
Acciarri, C; Khukhro, E; Shumyatsky, P
abstract

An element g of a group G is said to be right Engel if for every x∈ G there is a number n=n(g,x) such that [g,_n x]=1. We prove that if a profinite group G admits a coprime automorphism φ of prime order such that every fixed point of φ is a right Engel element, then G is locally nilpotent.

2019 - The generating graph of infinite abelian groups [Articolo su rivista]
Acciarri, C.; Lucchini, A.
abstract

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. 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 noncyclic abelian group, then Γ∗(G) is connected. Moreover, diam(Γ∗(G)) = 2 if the torsion subgroup of G is nontrivial and diam(Γ∗(G)) = ∞ otherwise. If F is the free group of rank 2, then Γ∗(F) is connected and we deduce from diam(Γ∗(Z × Z)) = ∞ that diam(Γ∗(F)) = ∞

2018 - Profinite groups and centralizers of coprime automorphisms whose elements are Engel [Articolo su rivista]
Acciarri, C; da Silveira, D S
abstract

Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^r$ with $rgeq2$ acting on a finite $q'$-group $G$. We show that if all elements in $gamma_{r-1}(C_G(a))$ are $n$-Engel in $G$ for any $ain A^#$, then $gamma_{r-1}(G)$ is $k$-Engel for some ${n,q,r}$-bounded number $k$, and if, for some integer $d$ such that $2^dleq r-1$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $G$ for any $ain A^#$, then the $d$th derived group $G^{(d)}$ is $k$-Engel for some ${n,q,r}$-bounded number $k$. Assuming $rgeq 3$ we prove that if all elements in $gamma_{r-2}(C_G(a))$ are $n$-Engel in $C_G(a)$ for any $ain A^#$, then $gamma_{r-2}(G)$ is $k$-Engel for some ${n,q,r}$-bounded number $k$, and if, for some integer $d$ such that $2^dleq r-2$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $C_G(a)$ for any $ain A^#,$ then the $d$th derived group $G^{(d)}$ is $k$-Engel for some ${n,q,r}$-bounded number $k$. Analogous (non-quantitative) results for profinite groups are also obtained.

2017 - Commutators and commutator subgroups in profinite groups [Articolo su rivista]
Acciarri, C; Shumyatsky, P
abstract

Let G be a profinite group. We prove that the commutator subgroup G′ is finite-by-procyclic if and only if the set of all commutators of G is contained in a union of countably many procyclic subgroups.

2017 - Coverings of commutators in profinite groups [Articolo su rivista]
Acciarri, C; Shumaytsky, P
abstract

Let w be a group-word. Suppose that the set of all w-values in a profinite group G is contained in a union of countably many subgroups. It is natural to ask in what way the structure of the verbal subgroup w(G) depends on the properties of the covering subgroups. The present article is a survey of recent results related to that question. In particular we survey results on finite and countable coverings of word-values (mostly commutators) by procyclic, abelian, nilpotent, and soluble subgroups, as well as subgroups with finiteness conditions. The last section of the paper is devoted to relation of the described results with Hall’s problem on conciseness of group-words.

2017 - On groups with automorphisms whose fixed points are Engel [Articolo su rivista]
Acciarri, C; Shumyatsky, P; da Silveira, D S
abstract

We study finite and profinite groups admitting an action by an elementary abelian group under which the centralizers of automorphisms consist of Engel elements. In particular, we prove the following theorems. Let $q$ be a prime and $A$ an elementary abelian $q$-group of order $q^2$ acting coprimely on a profinite group $G$. Assume that all elements in $C_{G}(a)$ are Engel in $G$ for each $ain A^{#}$. Then $G$ is locally nilpotent. Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^3$ acting coprimely on a finite group $G$. Assume that for each $ain A^{#}$ every element of $C_{G}(a)$ is $n$-Engel in $C_{G}(a)$. Then the group $G$ is $k$-Engel for some ${n,q}$-bounded number $k$.

2016 - Profinite groups and the fixed points of coprime automorphisms [Articolo su rivista]
Acciarri, C; Shumyatsky, P
abstract

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∈A#. Then the group G is locally nilpotent.

2014 - Centralizers of coprime automorphisms of finite groups [Articolo su rivista]
Acciarri, C; Shumyatsky, P.
abstract

Let A be an elementary abelian group of order pk with k ≥ 3 acting on a finite p′-group G. The following results are proved. If γk−2(CG(a)) is nilpotent of class at most c for any a ∈ A#, then γk−2(G) is nilpotent and has {c, k, p}-bounded nilpotency class. If, for some integer d such that 2d + 2 ≤ k, the dth derived group of CG (a) is nilpotent of class at most c for any a ∈ A#, then the dth derived group G(d) is nilpotent and has {c, k, p}-bounded nilpotency class.

2014 - Conciseness of coprime commutators in finite groups [Articolo su rivista]
Acciarri, C; Shumyatsky, P; Thillaisundaram, A
abstract

Let $G$ be a finite group. We show that the order of the subgroup generated by coprime $gamma_k$-commutators (respectively $delta_k$-commutators) is bounded in terms of the size of the set of coprime $gamma_k$-commutators (respectively $delta_k$-commutators). This is in parallel with the classical theorem due to Turner-Smith that the words $gamma_k$ and $delta_k$ are concise.

2014 - On finite groups in which coprime commutators are covered by few cyclic subgroups [Articolo su rivista]
Acciarri, C; Shumyatsky, P
abstract

The coprime commutators γj∗ and δj∗ were recently introduced as a tool to study properties of finite groups that can be expressed in terms of commutators of elements of coprime orders. Every element of a finite group G is both a γ1∗-commutator and a δ0∗-commutator. Now let j&gt;=2 and let X be the set of all elements of G that are powers of γj-1∗ -commutators. An element g is a γj∗ -commutator if there exist a ∈ X and b ∈ G such that g = [a,b] and (|a|,|b|) = 1. For j&gt;=1 let Y be the set of all elements of G that are powers of δj-1∗ -commutators. An element g is a δj∗ -commutator if there exist a,b ∈ Y such that g = [a,b] and (|a|,|b|) = 1. The subgroups of G generated by all γj∗-commutators and all δj∗-commutators are denoted by γj∗(G) and δj∗(G), respectively. For every j &gt;=2 the subgroup γj∗(G) is precisely the last term γ∞(G) of the lower central series of G, while for every j&gt;=1 the subgroup δj∗(G) is precisely the last term of the lower central series of δj∗−1(G), that is, δj∗(G) = γ∞ (δj-1∗ (G)). In the present paper we prove that if G possesses m cyclic subgroups whose union contains all γj∗-commutators of G, then γj∗(G) contains a subgroup Δ, of m-bounded order, which is normal in G and has the property that γj∗(G)/Δ is cyclic. If j&gt;=2 and G possesses m cyclic subgroups whose union contains all δj∗-commutators of G, then the order of δj∗(G) is m-bounded.

2014 - On words that are concise in residually finite groups [Articolo su rivista]
Acciarri, C; Shumyatsky, P
abstract

A group-word w is called concise if whenever the set of w-values in a group G is finite it always follows that the verbal subgroup w(G) is finite. More generally, a word w is said to be concise in a class of groups X if whenever the set of w-values is finite for a group G in the class X, it always follows that w(G) is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w is a multilinear commutator and q is a prime-power, then the word w^q is indeed concise in the class of residually finite groups. Further, we show that in the case where w=γ_k the word w^q is boundedly concise in the class of residually finite groups. It remains unknown whether the word w^q is actually concise in the class of all groups.

2013 - Double automorphisms of graded Lie algebras [Articolo su rivista]
Acciarri, C; Shumyatsky, P
abstract

We introduce the concept of a double automorphism of an $A$-graded Lie algebra $L$. Roughly, this is an automorphism of $L$ which also induces an automorphism of the group $A$. It is clear that the set of all double automorphisms of $L$ forms a subgroup in $Aut, L$. In the present paper we prove several nilpotency criteria for a graded Lie algebra admitting a finite group of double automorphisms. One of the obtained results is as follows. Let $A$ be a torsion-free abelian group and $L$ an $A$-graded Lie algebra in which $[L,\underbrace{L_0,ldots,L_0}_{k}]=0$. Assume that $L$ admits a finite group of double automorphisms $H$ such that $C_A(h)=0$ for all nontrivial $hin H$ and $C_L(H)$ is nilpotent of class $c$. Then $L$ is nilpotent and the class of $L$ is bounded in terms of $|H|$, $k$ and $c$ only. We also give an application of our results to groups admitting a Frobenius group of automorphisms.

2013 - On profinite groups in which commutators are covered by finitely many subgroups [Articolo su rivista]
Acciarri, C; Shumyatsky, P
abstract

For a family of group words w we show that if G is a profinite group in which all w-values are contained in a union of finitely 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. If G contains finitely many subgroups G1 , G2 , . . . , Gs of finite exponent e whose union contains all γk-values in G, it is shown that γk(G) has finite (e,k,s)-bounded exponent. If G contains finitely many subgroups G1, G2, . . . , Gs of finite rank r whose union contains all γk-values, it is shown that γk(G) has finite (k,r,s)-bounded rank.

2012 - A focal subgroup theorem for outer commutator words [Articolo su rivista]
Acciarri, C; FERNANDEZ-ALCOBER, G; Shumyatsky, P
abstract

Let $G$ be a finite group of order $p^am$, where $p$ is a prime and $m$ is not divisible by $p$, and let $P$ be a Sylow $p$-subgroup of $G$. If $w$ is an outer commutator word, we prove that $Pcap w(G)$ is generated by the intersection of $P$ with the set of $m$th powers of all values of $w$ in $G$

2012 - Derived subgroups of fixed points in profinite groups [Articolo su rivista]
Acciarri, C; DE SOUZA LIMA, A; Shumyatsky, P
abstract

The main result of this paper is the following theorem. Let q be a prime and A be an elementary abelian group of order q3. Suppose that A acts as a coprime group of automorphisms on a profinite group G in such a manner that C_G(a)′ is periodic for each a ∈ A#. Then G′ is locally finite.

2012 - On verbal subgroups in finite and profinite groups [Articolo su rivista]
Acciarri, C; Shumyatsky, P.
abstract

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

2012 - Positive laws on generators in powerful pro-p groups [Relazione in Atti di Convegno]
Acciarri, C; FERNÁNDEZ-ALCOBER G., A
abstract

If G is a finitely generated powerful pro-p group satisfying a certain law v == 1, and if G can be generated by a normal subset T of finite width which satisfies a positive law, we prove that G is nilpotent. Furthermore, the nilpotency class of G can be bounded in terms of the prime p, the number of generators of G, the law v = 1, the width of T, and the degree of the positive law. The main interest of this result is the application to verbal subgroups: if G is a p-adic analytic pro-p group in which all values of a word w satisfy positive law, and if the verbal subgroup w(G) is powerful, then w(G) is nilpotent.

2011 - Fixed points of coprime operator groups [Articolo su rivista]
Acciarri, C; Shumyatsky, P
abstract

Let m be a positive integer and A an elementary abelian group of order qr with r⩾2 acting on a finite q′-group G. We show that if for some integer d such that d2⩽r−1 the dth derived group of CG(a) has exponent dividing m for any a∈A#, then G(d) has {m,q,r}-bounded exponent and if γr−1(CG(a)) has exponent dividing m for any a∈A#, then γr−1(G) has {m,q,r}-bounded exponent.

2010 - Positive laws on large sets of generators: counterexamples for infinitely generated groups [Articolo su rivista]
Acciarri, C; FERNÁNDEZ-ALCOBER G., A
abstract

Shumyatsky and the second author proved that if G is a finitely generated residually finite p-group satisfying a law, then, for almost all primes p, the fact that a normal and commutator-closed set of generators satisfies a positive law implies that the whole of G also satisfies a (possibly different) positive law. In this paper, we construct a counterexample showing that the hypothesis of finite generation of the group G cannot be dispensed with.