On residually finite groups and their generalizations (original) (raw)

On Groups of Automorphisms of Residually Finite Groups

Journal of Algebra, 2000

We show that certain properties of groups of automorphisms can be read off from the actions they induce on the finite characteristic quotients of their underlying group G. In particular, we obtain criteria for groups of automorphisms of a Ž. residually finite and soluble minimax-by-finite group G to be nilpotent or soluble. Ž. Moreover, we give explicit bounds on the class the derived length, resp. of such groups of automorphisms in terms of invariants of G. Finally, we consider similar questions when G is the free group of rank two.

Residually finite properties of groups / Muhammad Sufi Mohd Asri

2018

In this thesis, we shall study two stronger forms of residual finiteness, namely cyclic subgroup separability and weak potency in various generalized free products and HNN extensions. Among our results, we shall show that the generalized free products and HNN extensions where the amalgamated or associated subgroups are finite, or central, or infinite cyclic, or they are direct products of an infinite cyclic subgroup with a finite subgroup, or they are finite extensions of central subgroups, are again cyclic subgroup separable or weakly potent respectively. In order to prove our results, we shall prove a criterion each for the weak potency of generalized free products and HNN extensions, but we shall use previously established criterions for cyclic subgroup separability. Finally, we shall extend our results to tree products and fundamental groups of graphs of groups.

Factorizations of infinite soluble groups

Rocky Mountain Journal of Mathematics, 1977

Introduction. If the group G = AB is the product of two of its subgroups A and B, then G is said to have a factorization with factors A and B, and G is factorized by its subgroups A and B. The main problem about factorized groups is the following question: What can be said about the structure of the factorized group G = AB if the structure of its subgroups A and B is known?

Locally soluble groups with min-n

Journal of the Australian Mathematical Society, 1974

It was shown by Baer in [1] that every soluble group satisfying Min-n, the minimal condition for normal subgroups, is a torsion group. Examples of non-soluble locally soluble groups satisfying Min-n have been known for some time (see McLain [2]), and these examples too are periodic. This raises the question whether all locally soluble groups with Min-n are torsion groups. We prove here that this is not the case, by establishing the existence of non-trivial locally soluble torsion-free groups satisfying Min-n. Rather than exhibiting one such group G, we give a general method for constructing examples; the reader will then be able to see that a variety of additional conditions may be imposed on G. It will follow, for instance, that G may be a Hopf group whose normal subgroups are linearly ordered by inclusion and are all complemented in G; further, that the countable groups G with these properties fall into exactly isomorphism classes. Again, there are exactly isomorphism classes of c...

The Influence of S-Embedded Subgroups on the Structure of Finite Groups

2015

Let H be a subgroup of a group G. H is said to be S-embedded in G if G has a normal subgroup T such that HT is an S-permutable subgroup of G and H \ T HsG, where HsG denotes the subgroup generated by all those subgroups of H which are S-permutable in G. In this paper, we investigate the inuence of minimal S-embedded subgroups on the structure of nite groups. We determine the structure of nite groups with some minimal S- embedded subgroups. We also give some new characterizations of p-nilpotency of nite groups in terms of the S-embedding property. As applications, some previously known results are generalized. Keywords: Finite groups, S-embedded subgroups, the generalized Fitting subgroups, soluble groups, p-nilpotent groups. MSC(2010): Primary: 20D10; Secondary: 20D15, 20D20, 20D25.

The non-abelian tensor square of residually finite groups

Monatshefte für Mathematik, 2016

Let m, n be positive integers and p a prime. We denote by ν(G) an extension of the non-abelian tensor square G ⊗ G by G × G. We prove that if G is a residually finite group satisfying some non-trivial identity f ≡ 1 and for every x, y ∈ G there exists a p-power q = q(x, y) such that [x, y ϕ ] q = 1, then the derived subgroup ν(G) ′ is locally finite (Theorem A). Moreover, we show that if G is a residually finite group in which for every x, y ∈ G there exists a p-power q = q(x, y) dividing p m such that [x, y ϕ ] q is left n-Engel, then the non-abelian tensor square G ⊗ G is locally virtually nilpotent (Theorem B).