arXiv subjects

Eloisa Detomi

Publications and source records attributed to Eloisa Detomi.

At least 19 recordsLinked to original sources

arXiv2026

Finite groups with high commuting probability for Sylow subgroups

Given two subsets $X,Y$ of a finite group $G$, we write $\Pr(X,Y)$ for the probability that random elements $x \in X$ and $y \in Y$ commute. If $X,Y$ are subgroups, we denote by $\Pr^*(X,Y)$ the maximum real number $ε$ with the property that for every pair of distinct primes $p\inπ(X)$ and $q\inπ(Y)$ there is a Sylow $p$-subgroup $P$ of $X$ and a Sylow $q$-subgroup $Q$ of $Y$ such that $\Pr(P,Q) \geq ε$. In this paper we handle, among other things, finite groups $G$ with high probabilities $\Pr^*(T,G)$, where $T$ is either a term of the lower central series of $G$ or the generalized Fitting subgroup $F_i^*(G)$. Our main results show that the structure of such groups is similar, in some precise sense, to that of nilpotent groups.

math.GR↗

arXiv2025

Finite groups, commuting probability, and coprime automorphisms

Given two subgroups $H,K$ of a finite group $G$, the probability that a pair of random elements from $H$ and $K$ commutes is denoted by $Pr(H,K)$. Suppose that a finite group $G$ admits a group of coprime automorphisms $A$ and let $ε>0$. We show that, if for any distinct primes $p,q\inπ(G)$ there is an $A$-invariant Sylow $p$-subgroup $P$ and an $A$-invariant Sylow $q$-subgroup $Q$ of $G$ for which $Pr([P,A],[Q,A])\geε$, then $F_2([G,A])$ has $ε$-bounded index in $[G,A]$ (Theorem 1.2). Here $F_2(K)$ stands for the second term of the upper Fitting seris of a group $K$. We also show that, if $G=[G,A]$ and for any prime $p$ dividing the order of $G$ there is an $A$-invariant Sylow $p$-subgroup $P$ such that $\Pr([P,A], [P,A]^x)\geqε$ for all $x\in G$, then $G$ is bounded-by-abelian-by-bounded (Theorem 1.4).

math.GR↗

arXiv2025

Commuting probability for conjugate subgroups of a finite group

Given two subgroups H,K of a finite group G, the probability that a pair of random elements from H and K commutes is denoted by \pr(H,K). We address the following question. Let P be a p-subgroup of a finite group G and assume that \pr(P,P^x)\geq\e>0 for every x\in G. Is the order of P modulo O_p(G) bounded in terms of e only? With respect to this question, we establish several positive results but show that in general the answer is negative. In particular, we prove that if the composition factors of G which are isomorphic to simple groups of Lie type in characteristic p, have Lie rank at most n, then the order of P modulo O_p(G) is bounded in terms of n and e only. If P is a Sylow p-subgroup of G, then the order of P modulo O_p(G) is bounded in terms e only. Some other results of similar flavour are established. We also show that if \pr(P_1,P_2)>0 for every two Sylow p-subgroups P_1,P_2 of a profinite group G, then O_{p,p'}(G) is open in G.

math.GR↗

arXiv2025

Groups with a covering condition on commutators

Given a group G and positive integers k,n, we let B_n=B_n(G) denote the set of all elements x in G such that |x^G|\leq n, and we say that G satisfies the (k,n)-covering condition for commutators if there is a subset S in G such that |S|\leq k and all commutators of G are contained in the product SB_n. The importance of groups satisfying this condition was revealed in the recent study of probabilistically nilpotent finite groups of class two. The main result obtained in this paper is the following theorem. Let G be a group satisfying the (k,n)-covering condition for commutators. Then G' contains a characteristic subgroup B such that [G':B] and |B'| are both (k,n)-bounded. This extends several earlier results of similar flavour.

math.GR↗

arXiv2024

Commuting probability for the Sylow subgroups of a profinite group

Given two subgroups $H,K$ of a compact group $G$, the probability that a random element of $H$ commutes with a random element of $K$ is denoted by $Pr(H,K)$. We show that if $G$ is a profinite group containing a Sylow $2$-subgroup $P$, a Sylow $3$-subgroup $Q_3$ and a Sylow $5$-subgroup $Q_5$ such that $Pr(P,Q_3)$ and $Pr(P,Q_5)$ are both positive, then $G$ is virtually prosoluble (Theorem 1.1). Furthermore, if $G$ is a prosoluble group in which for every subset $π\subseteqπ(G)$ there is a Hall $π$-subgroup $H_π$ and a Hall $π'$-subgroup $H_{π'}$ such that $Pr(H_π,H_{π'})>0$, then $G$ is virtually pronilpotent (Theorem 1.2).

math.GR↗

arXiv2024

Commuting probability for approximate subgroups of a finite group

For subsets X,Y of a finite group G, we write Pr(X,Y) for the probability that two random elements x in X and y in Y commute. This paper addresses the relation between the structure of an approximate subgroup A of G and the probabilities Pr(A,G) and Pr(A,A).

math.GR↗

arXiv2024

On groups with BFC-covered word values

For a group G and a positive integer n write B_n(G) = {x \in G : |x^G | \le n}. If s is a positive integer and w is a group word, say that G satisfies the (n,s)-covering condition with respect to the word w if there exists a subset S of G such that |S| \le s and all w-values of G are contained in B_n(G)S. In a natural way, this condition emerged in the study of probabilistically nilpotent groups of class two. In this paper we obtain the following results. Let w be a multilinear commutator word on k variables and let G be a group satisfying the (n,s)-covering condition with respect to the word w. Then G has a soluble subgroup T such that the index [G : T] and the derived length of T are both (k,n,s)-bounded. Let G be a group satisfying the (n,s)-covering condition with respect to the word γ_k. Then (1) γ_{2k-1}(G) has a subgroup $T$ such that the index [γ_{2k-1}(G) : T] and |T'| are both (k,n,s)-bounded; and (2) G has a nilpotent subgroup U such that the index [G : U] and the nilpotency class of U are both (k,n,s)-bounded.

math.GR↗

arXiv2023

Commuting probability for the Sylow subgroups of a finite group

For subsets $X,Y$ of a finite group $G$, let $Pr(X,Y)$ denote the probability that two random elements $x\in X$ and $y\in Y$ commute. Obviously, a finite group $G$ is nilpotent if and only if $Pr(P,Q)=1$ whenever $P$ and $Q$ are Sylow subgroups of $G$ of coprime orders. Suppose that $G$ is a finite group in which for any distinct primes $p,q\inπ(G)$ there is a Sylow $p$-subgroup $P$ and a Sylow $q$-subgroup $Q$ of $G$ such that $Pr(P,Q) \ge ε$. We show that $F_2(G)$ has $ε$-bounded index in $G$. If $G$ is a finite soluble group in which for any prime $p\inπ(G)$ there is a Sylow $p$-subgroup $P$ and a Hall $p'$-subgroup $H$ such that $Pr(P,H)\ge ε$, then $F(G)$ has $ε$-bounded index in $G$. Moreover, we establish criteria for nilpotency and solubility of $G$ such as: If for any primes $p,q\inπ(G)$ the group $G$ has a Sylow $p$-subgroup $P$ and a Sylow $q$-subgroup $Q$ with $Pr(P,Q)>2/3$, then $G$ is nilpotent. If for any primes $p,q\inπ(G)$ the group $G$ has a Sylow $p$-subgroup $P$ and a Sylow $q$-subgroup $Q$ with $Pr(P,Q)>2/5$, then $G$ is soluble.

math.GR↗

arXiv2023

Probabilistic properties of profinite groups

Let $\mathfrak C$ be a class of finite groups which is closed for subgroups, quotients and direct products. Given a profinite group $G$ and an element $x\in G$, we denote by $P_{\mathfrak{C}}(x,G)$ the probability that $x$ and a randomly chosen element of $G$ generate a pro-${\mathfrak C}$ subgroup. We say that a profinite group $G$ is $\mathfrak C$-positive if $P_{\mathfrak{C}}(x,G)>0$ for all $x \in G.$ %Moreover we say that $G$ is $\mathfrak C$-bounded-positive if there exists a positive constant $η$ such that $P_{\mathfrak{C}}(x,G)>η$ for all $x \in G.$ We establish several equivalent conditions for a profinite group to be $\mathfrak C$-positive when $\mathfrak C$ is the class of finite soluble groups or of finite nilpotent groups. In particular, for the above classes, the profinite $\mathfrak C$-positive groups are virtually prosoluble (resp., virtually nilpotent). We also draw some consequences on the prosoluble (resp. pronilpotent) graph of a profinite group.

math.GR↗

arXiv2022

Commutators, centralizers, and strong conciseness in profinite groups

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. We take interest in profinite groups with restricted centralizers of uniform commutators, that is, elements of the form $[x_1,\dots,x_k]$, where $π(x_1)=π(x_2)=\dots=π(x_k)$. Here $π(x)$ denotes the set of prime divisors of the order of $x\in G$. It is shown that such a group necessarily has an open nilpotent subgroup. We use this result to deduce that $γ_k(G)$ is finite if and only if the cardinality of the set of uniform $k$-step commutators in $G$ is less than $2^{\aleph_0}$

math.GR↗

arXiv2022

Strongly generating elements in finite and profinite groups

Given a finite group $G$ and an element $g\in G$, we may compare the expected number $e(G)$ of elements needed to generate $G$ and the expected number $e(G,g)$ of elements of $G$ needed to generate $G$ together with $g.$ We address the following question: how large can the difference $e(G)-e(G,g)$ be?

math.GR↗

arXiv2022

Centralizers of commutators in finite groups

Let $G$ be a finite group. A coprime commutator in $G$ is any element that can be written as a commutator $[x,y]$ for suitable $x,y\in G$ such that $π(x)\capπ(y)=\emptyset$. Here $π(g)$ denotes the set of prime divisors of the order of the element $g\in G$. An anti-coprime commutator is an element that can be written as a commutator $[x,y]$, where $π(x)=π(y)$. The main results of the paper are as follows. -- If $|x^G|\leq n$ whenever $x$ is a coprime commutator, then $G$ has a nilpotent subgroup of $n$-bounded index. -- If $|x^G|\leq n$ for every anti-coprime commutator $x\in G$, then $G$ has a subgroup $H$ of nilpotency class at most $4$ such that $[G : H]$ and $|γ_4 (H)|$ are both $n$-bounded. We also consider finite groups in which the centralizers of coprime, or anti-coprime, commutators are of bounded order.

math.GR↗

arXiv2022

The Engel graph of a finite group

For a finite group $G,$ we investigate the direct graph $Γ(G),$ whose vertices are the non-hypercentral elements of $G$ and where there is an edge $x\mapsto y$ if and only if $[x,_ny]=1$ for some $n \in \mathbb N.$ We prove that $Γ(G)$ is always weakly connected and is strongly connected if $G/Z_{\infty}(G)$ is neither Frobenius nor almost simple.

math.GR↗

arXiv2021

On the rank of a verbal subgroup of a finite group

We show that if $w$ is a multilinear commutator word and $G$ a finite group in which every metanilpotent subgroup generated by $w$-values is of rank at most $r$, then the rank of the verbal subgroup $w(G)$ is bounded in terms of $r$ and $w$ only. In the case where $G$ is soluble we obtain a better result -- if $G$ is a finite soluble group in which every nilpotent subgroup generated by $w$-values is of rank at most $r$, then the rank of $w(G)$ is at most $r+1$.

math.GR↗

arXiv2021

On the commuting probability for subgroups of a finite group

Let $K$ be a subgroup of a finite group $G$. The probability that an element of $G$ commutes with an element of $K$ is denoted by $Pr(K,G)$. Assume that $Pr(K,G)\geqε$ for some fixed $ε>0$. We show that there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indexes $[G:T]$ and $[K:B]$ and the order of the commutator subgroup $[T,B]$ are $ε$-bounded. This extends the well known theorem, due to P. M. Neumann, that covers the case where $K=G$. We deduce a number of corollaries of this result. A typical application is that if $K$ is the generalized Fitting subgroup $F^*(G)$ then $G$ has a class-2-nilpotent normal subgroup $R$ such that both the index $[G:R]$ and the order of the commutator subgroup $[R,R]$ are $ε$-bounded. In the same spirit we consider the cases where $K$ is a term of the lower central series of $G$, or a Sylow subgroup, etc.

math.GR↗

arXiv2021

Strong conciseness of coprime and anti-coprime commutators

A coprime commutator in a profinite group $G$ is an element of the form $[x,y]$, where $x$ and $y$ have coprime order and an anti-coprime commutator is a commutator $[x,y]$ such that the orders of $x$ and $y$ are divisible by the same primes. In the present paper we establish that a profinite group $G$ is finite-by-pronilpotent if the cardinality of the set of coprime commutators in $G$ is less than $2^{\aleph_0}$. Moreover, a profinite group $G$ has finite commutator subgroup $G'$ if the cardinality of the set of anti-coprime commutators in $G$ is less than $2^{\aleph_0}$.

math.GR↗

arXiv2020

Profinite groups in which the probabilistic zeta function has no negative coefficients

To a finitely generated profinite group $G$, a formal Dirichlet series $P_G(s)=\sum_{n \in \mathbb N} {a_n(G)}/{n^s}$ is associated, where $a_n(G)=\sum_{|G:H|=n}μ(H, G)$ and $μ(H,G)$ denotes the Möbius function of the lattice of open subgroups of $G.$ Its formal inverse $P_G^{-1}(s)$ is the probabilistic zeta function of $G$. When $G$ is prosoluble, every coefficient of $(P_G(s))^{-1}$ is nonnegative. In this paper we discuss the general case and we produce % existence of a non-prosoluble example and We construct a non-prosoluble finitely generated group $G$ with the same property.

math.GR↗

arXiv2020

Strong conciseness in profinite groups

A group word $w$ is said to be strongly concise in a class $\mathcal{C}$ of profinite groups if, for every group $G$ in $\mathcal{C}$ such that $w$ takes less than $2^{\aleph_0}$ values in $G$, the verbal subgroup $w(G)$ is finite. Detomi, Morigi and Shumyatsky established that multilinear commutator words -- and the particular words $x^2$ and $[x^2,y]$ -- have the property that the corresponding verbal subgroup is finite in a profinite group $G$ whenever the word takes at most countably many values in $G$. They conjectured that, in fact, this should be true for every word. In particular, their conjecture included as open cases power words and Engel words. In the present paper, we take a new approach via parametrised words that leads to stronger results. First we prove that multilinear commutator words are strongly concise in the class of all profinite groups. Then we establish that every group word is strongly concise in the class of nilpotent profinite groups. From this we deduce, for instance, that, if $w$ is one of the group words $x^2$, $x^3$, $x^6$, $[x^3,y]$ or $[x,y,y]$, then $w$ is strongly concise in the class of all profinite groups. Indeed, the same conclusion can be reached for all words of the infinite families $[x^m,z_1,\ldots,z_r]$ and $[x,y,y,z_1,\ldots,z_r]$, where $m \in \{2,3\}$ and $r \ge 1$.

math.GR↗