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Pavel Shumyatsky

Publications and source records attributed to Pavel Shumyatsky.

At least 37 records · Page 2Linked to original sources
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

Compact groups in which commutators have finite right Engel sinks

A right Engel sink of an element $g$ of a group $G$ is a subset containing all sufficiently long commutators $[...[[g,x],x],\dots ,x]$. We prove that if $G$ is a compact group in which, for some $k$, every commutator $[...[g_1,g_2],\dots ,g_k]$ has a finite right Engel sink, then $G$ has a locally nilpotent open subgroup. If in addition, for some positive integer $m$, every commutator $[...[g_1,g_2],\dots ,g_k]$ has a right Engel sink of cardinality at most $m$, then $G$ has a locally nilpotent subgroup of finite index bounded in terms of $m$ only.

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

Profinite groups with soluble centralisers

We show that a profinite group, in which the centralisers of non-trivial elements are metabelian, is either virtually pro-$p$ or virtually soluble of derived length at most 4. We furthermore show that a prosoluble group, in which the centralisers of non-trivial elements are soluble of bounded derived length, is either soluble or virtually pro-$p$.

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↗
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 profinite groups admitting a word with only few values

A group-word $w$ is called concise if the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values in a group $G$. It is known that there are words that are not concise. The problem whether every word is concise in the class of profinite groups remains wide open. Moreover, there is a conjecture that every word $w$ is strongly concise in profinite groups, that is, $w(G)$ is finite whenever $G$ is a profinite group in which $w$ takes less than $2^{\aleph_0}$ values. In this paper we show that if the word $w$ takes less than $2^{\aleph_0}$ values in a profinite group $G$ then $w(w(G))$ is finite.

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

Finite groups in which every commutator has prime power order

Finite groups in which every element has prime power order (EPPO-groups) are nowadays fairly well understood. For instance, if $G$ is a soluble EPPO-group, then the Fitting height of $G$ is at most 3 and $|π(G)|\leqslant 2$ (Higman, 1957). Moreover, Suzuki showed that if $G$ is insoluble, then the soluble radical of $G$ is a 2-group and there are exactly eight nonabelian simple EPPO-groups. In the present work we concentrate on finite groups in which every commutator has prime power order (CPPO-groups). Roughly, we show that if $G$ is a CPPO-group, then the structure of $G'$ is similar to that of an EPPO-group. In particular, we show that the Fitting height of a soluble CPPO-group is at most 3 and $|π(G')|\leqslant 3$. Moreover, if $G$ is insoluble, then $R(G')$ is a 2-group and $G'/R(G')$ is isomorphic to a simple EPPO-group.

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

On conciseness of the word in Olshanskii's example

A group-word $w$ is called concise if the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values in a group $G$. It is known that there are words that are not concise. In particular, Olshanskii gave an example of such a word, which we denote by $w_o$. The problem whether every word is concise in the class of residually finite groups remains wide open. In this note we observe that $w_o$ is concise in residually finite groups. Moreover, we show that $w_o$ is strongly concise in profinite groups, that is, $w_o(G)$ is finite whenever $G$ is a profinite group in which $w_o$ takes less than $2^{\aleph_0}$ values.

math.GR↗
arXiv2023

Strong conciseness of coprime commutators in profinite groups

Let $G$ be a profinite group. The coprime commutators $γ_j^*$ and $δ_j^*$ are defined as follows. Every element of $G$ is both a $γ_1^*$-value and a $δ_0^*$-value. For $j\geq 2$, let $X$ be the set of all elements of $G$ that are powers of $γ_{j-1}^*$-values. An element $a$ is a $γ_j^*$-value if there exist $x\in X$ and $g\in G$ such that $a=[x,g]$ and $(|x|,|g|)=1$. For $j\geq 1$, let $Y$ be the set of all elements of $G$ that are powers of $δ_{j-1}^*$-values. The element $a$ is a $δ_j^*$-value if there exist $x,y\in Y$ such that $a=[x,y]$ and $(|x|,|y|)=1$. In this paper we establish the following results. A profinite group $G$ is finite-by-pronilpotent if and only if there is $k$ such that the set of $γ_k^*$-values in $G$ has cardinality less than $2^{\aleph_0}$. A profinite group $G$ is finite-by-(prosoluble of Fitting height at most $k$) if and only if there is $k$ such that the set of $δ_k^*$-values in $G$ has cardinality less than $2^{\aleph_0}$.

math.GR↗
arXiv2023

Commuting and product-zero probability in finite rings

Let cp(R) be the probability that two random elements of a finite ring R commute and zp(R) the probability that the product of two random elements in R is zero. We show that if cp(R)=e, then there exists a Lie-ideal D in the Lie-ring (R,[.,.]) with e-bounded index and with [D,D] of e-bounded order. If zp(R)=e, then there exists an ideal D in R with e-bounded index and D^2 of e-bounded order. These results are analogous to the well-known theorem of P. Neumann on the commuting probability in finite groups.

math.RA↗
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↗
arXiv2023

Compact groups with high commuting probability of monothetic subgroups

If $H$ is a subgroup of a compact group $G$, the probability that a random element of $H$ commutes with a random element of $G$ is denoted by $Pr(H,G)$. Let $\langle g\rangle$ stand for the monothetic subgroup generated by an element $g\in G$ and let $K$ be a subgroup of $G$. We prove that $Pr(\langle x\rangle,G)>0$ for any $x\in K$ if and only if $G$ has an open normal subgroup $T$ such that $K/C_K(T)$ is torsion. In particular, $Pr(\langle x\rangle,G)>0$ for any $x\in G$ if and only if $G$ is virtually central-by-torsion, that is, there is an open normal subgroup $T$ such that $G/Z(T)$ is torsion. We also deduce a number of corollaries of this result.

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

Compact groups with probabilistically central monothetic subgroups

If $K$ is a closed subgroup of a compact group $G$, the probability that randomly chosen pair of elements from $K$ and $G$ commute is denoted by $Pr(K,G)$. Say that a subgroup $K\leq G$ is $ε$-central in $G$ if $Pr(\langle g \rangle,G)\geq ε$ for any $g$ in $K$. Here $\langle g \rangle$ denotes the monothetic subgroup generated by $g\in G$. Our main result is that if $K$ is $ε$-central in $G$, then there is an $ε$-bounded number $e$ and a normal subgroup $T\leq G$ such that the index $[G:T]$ and the order of the commutator subgroup $[K^e,T]$ both are finite and $ε$-bounded. In particular, if $G$ is a compact group for which there is $ε>0$ such that $Pr(\langle g \rangle,G)\geq ε$ for any $g \in G$, then there is an $ε$-bounded number $e$ and a normal subgroup $T$ such that the index $[G:T]$ and the order of $[G^e,T]$ both are finite and $ε$-bounded.

math.GR↗