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Pavel Shumyatsky

Publications and source records attributed to Pavel Shumyatsky.

At least 109 records · Page 6Linked to original sources
arXiv2014

On rational and concise words

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 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. It is still an open problem whether every word is concise in the class of residually finite groups. A word w is rational if the number of solutions to the equation w=g is the same as the number of solutions to w=g^e for every finite group G and for every e relatively prime to |G|. We observe that any rational word is concise in the class of residually finite groups. Further we give a sufficient condition for rationality of a word. This is used to produce a new family of words that are concise in the class of residually finite groups.

math.GR↗
arXiv2014

Procyclic coverings of commutators in profinite groups

We consider profinite groups in which all commutators are contained in a union of finitely many procyclic subgroups. It is shown that if G is a profinite group in which all commutators are covered by m procyclic subgroups, then G possesses a finite characteristic subgroup M contained in G' such that the order of M is m-bounded and G'/M is procyclic. If G is a pro-p group such that all commutators in G are covered by m procyclic subgroups, then G' is either finite of m-bounded order or procyclic.

math.GR↗
arXiv2014

On finite groups in which coprime commutators are covered by few cyclic subgroups

The coprime commutators $γ_j^*$ and $δ_j^*$ 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. They are defined as follows. Let $G$ be a finite group. Every element of $G$ is both a $γ_1^*$-commutator and a $δ_0^*$-commutator. Now let $j\geq 2$ and let $X$ be the set of all elements of $G$ that are powers of $γ_{j-1}^*$-commutators. An element $g$ is a $γ_j^*$-commutator if there exist $a\in X$ and $b\in G$ such that $g=[a,b]$ and $(|a|,|b|)=1$. For $j\geq 1$ let $Y$ be the set of all elements of $G$ that are powers of $δ_{j-1}^*$-commutators. The element $g$ is a $δ_j^*$-commutator if there exist $a,b\in Y$ such that $g=[a,b]$ and $(|a|,|b|)=1$. The subgroups of $G$ generated by all $γ_j^*$-commutators and all $δ_j^*$-commutators are denoted by $γ_j^*(G)$ and $δ_j^*(G)$, respectively. For every $j\geq2$ the subgroup $γ_j^*(G)$ is precisely the last term of the lower central series of $G$ (which throughout the paper is denoted by $γ_\infty(G)$) while for every $j\geq1$ the subgroup $δ_j^*(G)$ is precisely the last term of the lower central series of $δ_{j-1}^*(G)$, that is, $δ_j^*(G)=γ_\infty(δ_{j-1}^*(G))$. In the present paper we prove that if $G$ possesses $m$ cyclic subgroups whose union contains all $γ_j^*$-commutators of $G$, then $γ_j^*(G)$ contains a subgroup $Δ$, of $m$-bounded order, which is normal in $G$ and has the property that $γ_{j}^{*}(G)/Δ$ is cyclic. If $j\geq2$ and $G$ possesses $m$ cyclic subgroups whose union contains all $δ_j^*$-commutators of $G$, then the order of $δ_j^*(G)$ is $m$-bounded.

math.GR↗
arXiv2013

On countable coverings of word values in profinite groups

We prove the following results. Let w be a multilinear commutator word. If G is a profinite group in which all w-values are contained in a union of countably many periodic subgroups, then the verbal subgroup w(G) is locally finite. If G is a profinite group in which all w-values are contained in a union of countably many subgroups of finite rank, then the verbal subgroup w(G) has finite rank as well. As a by-product of the techniques developed in the paper we also prove that if G is a virtually soluble profinite group in which all w-values have finite order, then w(G) is locally finite and has finite exponent.

math.GR↗
arXiv2013

Conciseness of coprime commutators in finite groups

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

math.GR↗
arXiv2013

Bounding the Exponent of a Verbal Subgroup

We deal with the following conjecture. If w is a group word and G is a finite group in which any nilpotent subgroup generated by w-values has exponent dividing e, then the exponent of the verbal subgroup w(G) is bounded in terms of e and w only. We show that this is true in the case where w is either the nth Engel word or the word [x^n,y_1,y_2,...,y_k] (Theorem A). Further, we show that for any positive integer e there exists a number k=k(e) such that if w is a word and G is a finite group in which any nilpotent subgroup generated by products of k values of the word w has exponent dividing e, then the exponent of the verbal subgroup w(G) is bounded in terms of e and w only (Theorem B).

math.GR↗
arXiv2012

On words that are concise in residually finite groups

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 $G\in X$, 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.

math.GR↗
arXiv2012

Coprime commutators in PSL(2,q)

We show that every element of PSL(2,q) is a commutator of elements of coprime orders. This is proved by showing first that in PSL(2,q) any two involutions are conjugate by an element of odd order.

math.GR↗
arXiv2012

On groups admitting a word whose values are Engel

Let m, n be positive integers, v a multilinear commutator word and w = v^m. We prove that if G is a residually finite group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent. We also examine the question whether this is true in the case where G is locally graded rather than residually finite. We answer the question affirmatively in the case where m = 1. Moreover, we show that if u is a non-commutator word and G is a locally graded group in which all u-values are n-Engel, then the verbal subgroup u(G) is locally nilpotent.

math.GR↗
arXiv2012

Commutators of elements of coprime orders in finite groups

This paper is an attempt to find out which properties of a finite group G can be expressed in terms of commutators of elements of coprime orders. A criterion of solubility of G in terms of such commutators is obtained. We also conjecture that every element of a nonabelian simple group is a commutator of elements of coprime orders and we confirm this conjecture for the alternating groups.

math.GR↗
arXiv2012

Double automorphisms of graded Lie algebras

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. We also give an application of our results to groups admitting a Frobenius group of automorphisms.

math.RA↗
arXiv2012

On the Exponent of a Verbal Subgroup in a Finite Group

Let w be a multilinear commutator word. We prove that if e is a positive integer and G is a finite group in which any nilpotent subgroup generated by w-values has exponent dividing e then the exponent of the corresponding verbal subgroup w(G) is bounded in terms of e and w only.

math.GR↗
arXiv2012

The ranks of central factor and commutator groups

The Schur Theorem says that if $G$ is a group whose center $Z(G)$ has finite index $n$, then the order of the derived group $G'$ is finite and bounded by a number depending only on $n$. In the present paper we show that if $G$ is a finite group such that $G/Z(G)$ has rank $r$, then the rank of $G'$ is $r$-bounded. We also show that a similar result holds for a large class of infinite groups.

math.GR↗
arXiv2011

A focal subgroup theorem for outer commutator words

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 $P\cap w(G)$ is generated by the intersection of $P$ with the set of $m$th powers of all values of $w$ in $G$

math.GR↗
arXiv2011

On profinite groups in which commutators are covered by finitely many subgroups

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 $w(G)$ 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 $G_1,G_2,...,G_s$ of finite exponent $e$ whose union contains all $γ_k$-values in $G$, it is shown that $γ_k(G)$ has finite $(e,k,s)$-bounded exponent. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite rank $r$ whose union contains all $γ_k$-values, it is shown that $γ_k(G)$ has finite $(k,r,s)$-bounded rank.

math.GR↗
arXiv2011

Centralizers of coprime automorphisms of finite groups

Let $A$ be an elementary abelian group of order $p^{k}$ with $k\geq 3$ acting on a finite $p'$-group $G$. The following results are proved. If $γ_{k-2}(C_{G}(a))$ is nilpotent of class at most $c$ for any $a\in A^{#}$, then $γ_{k-2}(G)$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. If, for some integer $d$ such that $2^{d}+2\leq k$, the $d$th derived group of $C_{G}(a)$ is nilpotent of class at most $c$ for any $a\in A^{#}$, then the $d$th derived group $G^{(d)}$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. Earlier this was known only in the case where $k\leq 4$.

math.GR↗
arXiv2010

Multilinear commutators in residually finite groups

Let w be a multilinear commutator and n a positive integer. Suppose that G is a residually finite group in which every product of at most 896 w-values has order dividing n. Then the verbal subgroup w(G) is locally finite.

math.GR↗