scholarly journals BFC-theorems for higher commutator subgroups

2018 ◽  
Vol 70 (3) ◽  
pp. 849-858 ◽  
Author(s):  
Eloisa Detomi ◽  
Marta Morigi ◽  
Pavel Shumyatsky
Keyword(s):  
Study Groups ◽  
Conjugacy Classes ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
Derived Subgroup ◽  
Bounded Size

Abstract A BFC-group is a group in which all conjugacy classes are finite with bounded size. In 1954, B. H. Neumann discovered that if G is a BFC-group then the derived group G′ is finite. Let w=w(x1,…,xn) be a multilinear commutator. We study groups in which the conjugacy classes containing w-values are finite of bounded order. Let G be a group and let w(G) be the verbal subgroup of G generated by all w-values. We prove that if |xG|≤m for every w-value x, then the derived subgroup of w(G) is finite of order bounded by a function of m and n. If |xw(G)|≤m for every w-value x, then [w(w(G)),w(G)] is finite of order bounded by a function of m and n.

scholarly journals ON GROUPS ADMITTING A WORD WHOSE VALUES ARE ENGEL

2013 ◽  
Vol 23 (01) ◽  
pp. 81-89 ◽  
Author(s):  
RAIMUNDO BASTOS ◽  
PAVEL SHUMYATSKY ◽  
ANTONIO TORTORA ◽  
MARIA TOTA
Keyword(s):  
Finite Group ◽  
Multilinear Commutator ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Verbal Subgroup ◽  
Locally Nilpotent ◽  
Locally Graded Group ◽  
Commutator Word ◽  
Positive Integers

Let m, n be positive integers, v a multilinear commutator word and w = vm. 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.

scholarly journals ON THE EXPONENT OF A VERBAL SUBGROUP IN A FINITE GROUP

2012 ◽  
Vol 93 (3) ◽  
pp. 325-332 ◽  
Author(s):  
PAVEL SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Nilpotent Subgroup ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
A Finite Group ◽  
Commutator Word

AbstractLet $w$ 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.

scholarly journals Profinite groups with restricted centralizers of commutators

2019 ◽  
Vol 150 (5) ◽  
pp. 2301-2321 ◽  
Author(s):  
Eloisa Detomi ◽  
Marta Morigi ◽  
Pavel Shumyatsky
Keyword(s):  
Finite Index ◽  
Profinite Group ◽  
Profinite Groups ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
Commutator Word

AbstractA group G has restricted centralizers if for each g in G the centralizer $C_G(g)$ either is finite or has finite index in G. A theorem of Shalev states that a profinite group with restricted centralizers is abelian-by-finite. In the present paper we handle profinite groups with restricted centralizers of word-values. We show that if w is a multilinear commutator word and G a profinite group with restricted centralizers of w-values, then the verbal subgroup w(G) is abelian-by-finite.

scholarly journals On locally graded groups with a word whose values are Engel

2015 ◽  
Vol 59 (2) ◽  
pp. 533-539 ◽  
Author(s):  
Pavel Shumyatsky ◽  
Antonio Tortora ◽  
Maria Tota
Keyword(s):  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
Locally Nilpotent ◽  
Locally Graded Group ◽  
Commutator Word ◽  
Locally Graded Groups ◽  
Positive Integers

AbstractLet m, n be positive integers, let υ be a multilinear commutator word and let w = υm. We prove that if G is a locally graded group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent.

scholarly journals ON THE RANK OF A VERBAL SUBGROUP OF A FINITE GROUP

2021 ◽  
pp. 1-15
Author(s):  
ELOISA DETOMI ◽  
MARTA MORIGI ◽  
PAVEL SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Soluble Group ◽  
Nilpotent Subgroup ◽  
Finite Soluble Group ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
A Finite Group ◽  
Commutator Word

Abstract 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 $r+1$ .

scholarly journals On finite groups in which commutators are covered by Engel subgroups

2019 ◽  
Vol 22 (6) ◽  
pp. 1049-1057
Author(s):  
Pavel Shumyatsky ◽  
Danilo Silveira
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
Bounded Number ◽  
A Finite Group ◽  
Commutator Word ◽  
Positive Integers

Abstract Let {m,n} be positive integers and w a multilinear commutator word. Assume that G is a finite group having subgroups {G_{1},\ldots,G_{m}} whose union contains all w-values in G. Assume further that all elements of the subgroups {G_{1},\ldots,G_{m}} are n-Engel in G. It is shown that the verbal subgroup {w(G)} is s-Engel for some {\{m,n,w\}} -bounded number s.

On Solvability of Groups with a Few Non-cyclic Subgroups

2016 ◽  
Vol 23 (01) ◽  
pp. 105-110 ◽  
Author(s):  
Mohammad Zarrin
Keyword(s):  
Finite Group ◽  
Sufficient Conditions ◽  
Study Groups ◽  
Conjugacy Classes ◽  
Cyclic Subgroups ◽  
A Finite Group

Li and Zhao studied groups with a few conjugacy classes of non-cyclic subgroups. In this paper we study groups with a few non-cyclic subgroups. In fact, among other things, we give some sufficient conditions on the number of non-cyclic subgroups of a finite group to be solvable.

scholarly journals WORDS AND PRONILPOTENT SUBGROUPS IN PROFINITE GROUPS

2014 ◽  
Vol 97 (3) ◽  
pp. 343-364 ◽  
Author(s):  
E. I. KHUKHRO ◽  
P. SHUMYATSKY
Keyword(s):  
Profinite Group ◽  
Profinite Groups ◽  
Locally Finite ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
Commutator Word

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}w$ be a multilinear commutator word, that is, a commutator of weight $n$ in $n$ different group variables. It is proved that if $G$ is a profinite group in which all pronilpotent subgroups generated by $w$-values are periodic, then the verbal subgroup $w(G)$ is locally finite.

scholarly journals Nilpotent-by-Černikov CC-groups

1992 ◽  
Vol 53 (1) ◽  
pp. 120-130 ◽  
Author(s):  
Javier Otal ◽  
Juan Manuel Peña
Keyword(s):  
Study Groups ◽  
Conjugacy Classes

AbstractIn this paper we study groups with Černikov conjugacy classes which are nilpotent-by-Černikov groups, giving full characterizations of them and applying the results obtained to some related areas.

Groups in which squares have boundedly many conjugates

2019 ◽  
Vol 22 (1) ◽  
pp. 133-136
Author(s):  
Gláucia Dierings ◽  
Pavel Shumyatsky
Keyword(s):  
Positive Integer ◽  
Conjugacy Class ◽  
Present Article ◽  
Conjugacy Classes ◽  
Famous Result ◽  
Bounded Size

Abstract Given a group G, we write {x^{G}} for the conjugacy class of G containing the element x. A famous result 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^{\prime}} is finite. Recently we showed that if {|x^{G}|\leq n} for any commutator x, then {|G^{\prime\prime}|} is finite and n-bounded. If {|x^{G^{\prime}}|\leq n} for any commutator x, then {|\gamma_{3}(G^{\prime})|} is finite and n-bounded. The present article deals with groups in which the conjugacy classes containing squares are finite with bounded size. The following theorem is proved. Let n be a positive integer, G a group and H the subgroup generated by all squares in G. If {|x^{H}|\leq n} for any square {x\in G} , then the order of {\gamma_{3}(H)} is finite and n-bounded.

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