140 research outputs found
On residually finite groups with Engel-like conditions
Let be positive integers. Suppose that is a residually finite group
in which for every element there exists a positive integer such that is -Engel. We show that is locally
virtually nilpotent. Further, let be a multilinear commutator and a
residually finite group in which for every product of at most -values
there exists a positive integer dividing such that is
-Engel. Then is locally virtually nilpotent.Comment: 9 page
A Sufficient Condition for Nilpotency of the Commutator Subgroup
Let be a finite group with the property that if are commutators of
coprime orders, then . We show that is nilpotent
On profinite groups with Engel-like conditions
Let be a profinite group in which for every element there exists
a natural number such that is Engel. We show that is locally
virtually nilpotent. Further, let be a prime and a finitely generated
profinite group in which for every -value there exists a
natural -power such that is Engel. We show that
is locally virtually nilpotent
Non-abelian tensor product of residually finite groups
Let and be groups that act compatibly on each other. We denote by
a certain extension of the non-abelian tensor product
by . Suppose that is residually finite and the subgroup satisfies some non-trivial
identity . We prove that if is a prime and every tensor has
-power order, then the non-abelian tensor product is locally
finite. Further, we show that if is a positive integer and every tensor is
left -Engel in , then the non-abelian tensor product is locally nilpotent. The content of this paper extend some results
concerning the non-abelian tensor square .Comment: Dedicated to Professor Antonio Paques on the occasion of his 70th
anniversary, S\~ao Paulo J. Math. Sci. (2017
Non-abelian tensor square of finite-by-nilpotent groups
Let be a group. We denote by an extension of the non-abelian
tensor square by . We prove that if is
finite-by-nilpotent, then the non-abelian tensor square is
finite-by-nilpotent. Moreover, is nilpotent-by-finite (Theorem A).
Also we characterize BFC-groups in terms of (Theorem B)
On finite groups with few automorphism orbits
Denote by the number of orbits of the action of on the
finite group . We prove that if is a finite nonsolvable group in which
, then is isomorphic to one of the groups
or . We also consider the case when and show that if is a nonsolvable finite group with ,
then either or there exists a characteristic elementary
abelian -subgroup of such that .Comment: accepted for publication in Communications in Algebr
The non-abelian tensor square of residually finite groups
Let be positive integers and a prime. We denote by an
extension of the non-abelian tensor square by . We
prove that if is a residually finite group satisfying some non-trivial
identity and for every there exists a -power
such that , then the derived subgroup
is locally finite (Theorem A). Moreover, we show that if is a
residually finite group in which for every there exists a -power
dividing such that is left -Engel, then
the non-abelian tensor square is locally virtually nilpotent
(Theorem B).Comment: 11 pages. arXiv admin note: substantial text overlap with
arXiv:1505.0446
A criterion for metanilpotency of a finite group
We prove that the th term of the lower central series of a finite group
is nilpotent if and only if for any -commutators
of coprime orders
FC-groups with finitely many automorphism orbits
Let be a group. The orbits of the natural action of on are
called "automorphism orbits" of , and the number of automorphism orbits of
is denoted by . In this paper we prove that if is an
FC-group with finitely many automorphism orbits, then the derived subgroup
is finite and admits a decomposition , where
is the torsion subgroup of and is a divisible characteristic subgroup
of . We also show that if is an infinite FC-group with , then either is soluble or , where
is an infinite abelian group with . Moreover, we describe the
structure of the infinite non-soluble FC-groups with at most eleven
automorphism orbits.Comment: Submitted to an internacional journa
Finite Groups with 6 or 7 Automorphism Orbits
Let be a group. The orbits of the natural action of \mbox{Aut}(G) on
are called "automorphism orbits" of , and the number of automorphism
orbits of is denoted by . In this paper the finite nonsolvable
groups with are classified - this solves a problem posed
by Markus Stroppel - and it is proved that there are infinitely many finite
nonsolvable groups with . Moreover it is proved that for a
given number there are only finitely many finite groups without
nontrivial abelian normal subgroups and such that ,
generalizing a result of Kohl
- …