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Pavel Shumyatsky

Publications and source records attributed to Pavel Shumyatsky.

At least 91 records · Page 5Linked to original sources
arXiv2017

Orderable groups with Engel-like conditions

Let x be an element of a group G. For a positive integer n let E_n(x) be the subgroup generated by all commutators [...[[y,x],x],...,x] over y in G, where x is repeated n times. There are several recent results showing that certain properties of groups with small subgroups E_n(x) are close to those of Engel groups. The present article deals with orderable groups in which, for some n, the subgroups E_n(x) are polycyclic. Let h,n be positive integers and G an orderable group in which E_n(x) is polycyclic with Hirsch length at most h for every x in G. It is proved that there are (h,n)-bounded numbers h* and c* such that G has a finitely generated normal nilpotent subgroup N with h(N)<h* and G/N nilpotent of class at most c*.

math.GR↗
arXiv2017

On just-infinite periodic locally soluble groups

We construct an uncountable family of periodic locally soluble groups which are hereditarily just infinite. We also show that the associated full C*-algebra C*(G) is just infinite for many groups $G$ in this family.

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↗
arXiv2017

On groups with automorphisms whose fixed points are Engel

We complete the study of 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 at least $q^2$ acting coprimely on a profinite group $G$. Assume that all elements in $C_{G}(a)$ are Engel in $G$ for each $a\in A^{\#}$. Then $G$ is locally nilpotent (Theorem B2). 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 $a\in A^{\#}$ 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 $k$ (Theorem A3).

math.GR↗
arXiv2016

Almost Engel linear groups

We prove that if a linear group $G$ is almost Engel, then $G$ is finite-by-hypercentral. If $G$ is almost nil, then $G$ is finite-by-nilpotent.

math.GR↗
arXiv2016

On finite groups with automorphisms whose fixed points are Engel

The main result of the paper is the following theorem. Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^2$. Suppose that $A$ acts coprimely on a finite group $G$ and assume that for each $a\in A^{\#}$ every element of $C_{G}(a)$ is $n$-Engel in $G$. Then the group $G$ is $k$-Engel for some $\{n,q\}$-bounded number $k$.

math.GR↗
arXiv2015

Coverings of commutators in profinite groups

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.

math.GR↗
arXiv2015

On conciseness of words in profinite groups

Let w be a group word. It is conjectured that if w has only countably many values in a profinite group G, then the verbal subgroup w(G) is finite. In the present paper we confirm the conjecture in the cases where w is a multilinear commutator word, or the word x^2, or the word [x^2,y].

math.GR↗
arXiv2015

On profinite groups with Engel-like conditions

Let $G$ be a profinite group in which for every element $x\in G$ there exists a natural number $q=q(x)$ such that $x^q$ is Engel. We show that $G$ is locally virtually nilpotent. Further, let $p$ be a prime and $G$ a finitely generated profinite group in which for every $γ_k$-value $x\in G$ there exists a natural $p$-power $q=q(x)$ such that $x^q$ is Engel. We show that $γ_k(G)$ is locally virtually nilpotent.

math.GR↗
arXiv2015

On the length of a finite group and of its 2-generator subgroups

The nonsoluble length $λ(G)$ of a finite group $G$ is defined as the minimum number of nonsoluble factors in a normal series of $G$ each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group $G$ is the least number $h=h^*(G)$ such that $F^*_h(G)=G$, where $F^*_1(G)=F^*(G)$ is the generalized Fitting subgroup, and $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_{i}(G))$. In the present paper we prove that if $λ(J)\leq k$ for every 2-generator subgroup $J$ of $G$, then $λ(G)\leq k$. It is conjectured that if $h^*(J)\leq k$ for every 2-generator subgroup $J$, then $h^*(G)\leq k$. We prove that if $h^*(\langle x,x^g\rangle)\leq k$ for all $x,g\in G$ such that $\langle x,x^g\rangle$ is soluble, then $h^*(G)$ is $k$-bounded.

math.GR↗
arXiv2015

On profinite groups with commutators covered by nilpotent subgroups

Let G be a profinite group. The following results are proved. The commutator subgroup G' is finite if and only if G is covered by countably many abelian subgroups. The group G is finite-by-nilpotent if and only if G is covered by countably many nilpotent subgroups. The main result is that the commutator subgroup G' is finite-by-nilpotent if and only if the set of commutators in G is covered by countably many nilpotent subgroups.

math.GR↗
arXiv2015

Nonsoluble Length Of Finite Groups with Commutators of Small Order

Let p be a prime. Every finite group G has a normal series each of whose quotients either is p-soluble or is a direct product of nonabelian simple groups of orders divisible by p. The non-p-soluble length of G is defined as the minimal number of non-p-soluble quotients in a series of this kind. We deal with the question whether, for a given prime p and a given proper group variety V, there is a bound for the non-p-soluble length of finite groups whose Sylow p-subgroups belong to V. Let the word w be a multilinear commutator. In the present paper we answer the question in the affirmative in the case where p is odd and the variety is the one of groups in which the w-values have orders dividing a fixed number.

math.GR↗