1,720,972 research outputs found
Coprime commutators in finite groups
No full textLet G be a finite group and let k≥2 We prove that the coprime subgroup γ*k(G) is nilpotent if and only if |xy|=|x||y| for any γ*k -commutators x,yεG of coprime orders (Theorem A). Moreover, we show that the coprime subgroup γ*k(G) is nilpotent if and only if |ab|=|a||b| for any powers of γ*k -commutators a,bεG of coprime orders (Theorem B)
A nilpotency criterion for some verbal subgroups
No full textThe word w = [xi1 , xi2 , : : : ; xik] is a simple commutator word if k ≤ 2; i1 ≠ i2 and ij i {1,...m} for some m>1. For a finite group G, we prove that if i1 ≠ ij for every j ≠ 1, then the verbal subgroup corresponding to is nilpotent if and only if |ab|=|a||b| for any w-values a,b iG 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
ADVANCES ON A CONSTRUCTION RELATED TO THE NON-ABELIAN TENSOR SQUARE OF A GROUP
No full textThis is a survey on a group construction in connection with the non-abelian tensor square of groups. We report on the developments obtained in the last decade emphasizing the results from a commutator point of view
BOUNDEDLY FINITE CONJUGACY CLASSES OF TENSORS
No full textLet n be a positive integer and let G be a group. We denote by nu(G) a certain extension of the non-abelian tensor square G circle times G by G x G. Set T-circle times(G) = {g circle times h vertical bar g, h is an element of G}. We prove that if the size of the conjugacy class vertical bar x(nu(G))vertical bar <= n for every x is an element of T-circle times(G), then the second derived subgroup nu(G)'' is finite with n-bounded order. Moreover, we obtain a sufficient condition for a group to be a BFC-group
A conjecture related to the nilpotency of groups with isomorphic non-commuting graphs
Full text linkIn this work we discuss whether the non-commuting graph of a finite group can determine its nilpotency. More precisely, Abdollahi, Akbari and Maimani conjectured that if G and H are finite groups with isomorphic non-commuting graphs and G is nilpotent, then H must be nilpotent as well (Conjecture 2). We characterize the structure of such an H when G is a finite AC-group, that is, a finite group in which all centralizers of non-central elements are abelian. As an application, we prove Conjecture 2 for finite AC-groups whenever |Z(G)|≥|Z(H)|
Groups with some families of complemented subgroups
No full textA subgroup H of a group G is said to be complemented in G if there exists a subgroup K of G such that G= HK and H∩ K= 1. We prove that, for a locally soluble group G, all cyclic subgroups are complemented if and only if it is the semidirect product of groups A=Dri∈IAi by B=Drj∈JBj, where all factors Ai and Bj are finite of prime order, and A has a set of maximal subgroups normal in G with trivial intersection. An analysis of the structure of periodic locally soluble groups of infinite rank shows, in particular, that if G is a periodic locally soluble group whose infinite rank subgroups are complemented, then every subgroup of G is complemented
A criterion for metanilpotency of a finite group
No full textWe prove that the kth term of the lower central series of a finite group G is nilpotent if and only if jabj D jajjbj for any k-commutators a; b 2 G of coprime orders
Upper bounds for the product of element orders of finite groups
Full text linkLet G be a finite group of order n, and denote by ρ(G) the product of element orders of G. The aim of this work is to provide some upper bounds for ρ(G) depending only on n and on its least prime divisor, when G belongs to some classes of non-cyclic groups
On generalized concise words
No full textThe study of verbal subgroups within a group is well known for being an effective tool to obtain structural information about a group. Therefore, conditions that allow the classification of words in a free group are of paramount importance. One of the most studied problems is to establish which words are concise, where a word w is said to be concise if the verbal subgroup w(G) is finite in each group G in which w takes only a finite number of values. The purpose of this article is to present some results, in which a hierarchy among words is introduced, generalizing the concept of concise word
On the structure of finite groups determined by the arithmetic and geometric means of element orders
Full text linkIn this paper we consider two functions related to the arithmetic and geometric means of element orders of a finite group, showing that certain lower bounds on such functions strongly affect the group structure. In particular, for every prime p, we prove a sufficient condition for a finite group to be p-nilpotent, that is, a group whose elements of (Formula presented.) -order form a normal subgroup. Moreover, we characterize finite cyclic groups with prescribed number of prime divisors
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