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.

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 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 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 Non-Abelian finite groups whose character sums are invariant but are not Cayley isomorphism

2019 ◽  
Vol 18 (01) ◽  
pp. 1950013
Author(s):  
Alireza Abdollahi ◽  
Maysam Zallaghi
Keyword(s):  
Finite Group ◽  
Cayley Graph ◽  
Finite Groups ◽  
Closed Subset ◽  
Cayley Graphs ◽  
Character Sums ◽  
Main Question ◽  
Vertex Set ◽  
A Finite Group ◽  
Positive Integers

Let [Formula: see text] be a group and [Formula: see text] an inverse closed subset of [Formula: see text]. By a Cayley graph [Formula: see text], we mean the graph whose vertex set is the set of elements of [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text]. A group [Formula: see text] is called a CI-group if [Formula: see text] for some inverse closed subsets [Formula: see text] and [Formula: see text] of [Formula: see text], then [Formula: see text] for some automorphism [Formula: see text] of [Formula: see text]. A finite group [Formula: see text] is called a BI-group if [Formula: see text] for some inverse closed subsets [Formula: see text] and [Formula: see text] of [Formula: see text], then [Formula: see text] for all positive integers [Formula: see text], where [Formula: see text] denotes the set [Formula: see text]. It was asked by László Babai [Spectra of Cayley graphs, J. Combin. Theory Ser. B 27 (1979) 180–189] if every finite group is a BI-group; various examples of finite non-BI-groups are presented in [A. Abdollahi and M. Zallaghi, Character sums of Cayley graph, Comm. Algebra 43(12) (2015) 5159–5167]. It is noted in the latter paper that every finite CI-group is a BI-group and all abelian finite groups are BI-groups. However, it is known that there are finite abelian non-CI-groups. Existence of a finite non-abelian BI-group which is not a CI-group is the main question which we study here. We find two non-abelian BI-groups of orders 20 and 42 which are not CI-groups. We also list all BI-groups of orders up to 30.

scholarly journals A NILPOTENCY CRITERION FOR SOME VERBAL SUBGROUPS

2019 ◽  
Vol 100 (2) ◽  
pp. 281-289
Author(s):  
CARMINE MONETTA ◽  
ANTONIO TORTORA
Keyword(s):  
Finite Group ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Verbal Subgroup ◽  
A Finite Group ◽  
Commutator Word ◽  
Verbal Subgroups

The word $w=[x_{i_{1}},x_{i_{2}},\ldots ,x_{i_{k}}]$ is a simple commutator word if $k\geq 2,i_{1}\neq i_{2}$ and $i_{j}\in \{1,\ldots ,m\}$ for some $m>1$. For a finite group $G$, we prove that if $i_{1}\neq i_{j}$ for every $j\neq 1$, then the verbal subgroup corresponding to $w$ is nilpotent if and only if $|ab|=|a||b|$ for any $w$-values $a,b\in G$ of coprime orders. We also extend the result to a residually finite group $G$, provided that the set of all $w$-values in $G$ is finite.

scholarly journals Ensuring commutativity of finite groups

2001 ◽  
Vol 71 (2) ◽  
pp. 233-234 ◽  
Author(s):  
B. H. Neumann
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group ◽  
Positive Integers

AbstractComments are made on the following question. Let m, n be positive integers and g a finite group. Suppose that for all choices of a subset of cardinality m and of a subset of cardinality n in g some member of the first commutes with some member of the second. Under what conditions on m, n is the group abelian?

scholarly journals ON VERBAL SUBGROUPS IN RESIDUALLY FINITE GROUPS

2011 ◽  
Vol 84 (1) ◽  
pp. 159-170 ◽  
Author(s):  
JHONE CALDEIRA ◽  
PAVEL SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Locally Finite ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Verbal Subgroup ◽  
Residually Finite Groups ◽  
Positive Integers ◽  
Verbal Subgroups

AbstractThe following theorem is proved. Let m, k and n be positive integers. There exists a number η=η(m,k,n) depending only on m, k and n such that if G is any residually finite group satisfying the condition that the product of any η commutators of the form [xm,y1,…,yk ] is of order dividing n, then the verbal subgroup of G corresponding to the word w=[xm,y1,…,yk ] is locally finite.

NOTES ON FINITE SIMPLE GROUPS WHOSE ORDERS HAVE THREE OR FOUR PRIME DIVISORS

2009 ◽  
Vol 08 (03) ◽  
pp. 389-399 ◽  
Author(s):  
LIANGCAI ZHANG ◽  
GUIYUN CHEN ◽  
SHUNMIN CHEN ◽  
XUEFENG LIU
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
Order Component ◽  
A Finite Group ◽  
Order Components ◽  
Positive Integers

Based on the prime graph of a finite group, its order can be divided into a product of some co-prime positive integers. These integers are called order components of this group. If there exist exactly k nonisomorphic finite groups with the same set of order components of a given finite group, we say that it is a k-recognizable group by its order component(s). In the present paper, we obtain that all finite simple Kn-groups (n = 3, 4) except U4(2) and A10can be uniquely determined by their order components. Moreover, U4(2) is 2-recognizable and A10is k-recognizable, where k denotes the number of all nonisomorphic classes of groups with the same order as A10. As a consequence of this result we can obtain some interesting corollaries.

scholarly journals Fusion systems and group actions with abelian isotropy subgroups

2013 ◽  
Vol 56 (3) ◽  
pp. 873-886 ◽  
Author(s):  
Özgün Ünlü ◽  
Ergün Yalçin
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Finite Solvable Group ◽  
Recursive Method ◽  
Fusion Systems ◽  
Product Of Spheres ◽  
A Finite Group ◽  
Positive Integers

AbstractWe prove that if a finite group G acts smoothly on a manifold M such that all the isotropy subgroups are abelian groups with rank ≤ k, then G acts freely and smoothly on M × $\mathbb{S}^{n_1}\$ × … × $\mathbb{S}^{n_k}$ for some positive integers n1, …, nk. We construct these actions using a recursive method, introduced in an earlier paper, that involves abstract fusion systems on finite groups. As another application of this method, we prove that every finite solvable group acts freely and smoothly on some product of spheres, with trivial action on homology.

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