Large Characteristic Subgroups Research Articles

Large Characteristic Subgroups with Restricted Conjugacy Classes

It is well-known that every virtually abelian group contains an abelian characteristic subgroup of finite index. We shall say that a group class $$\mathfrak {X}$$ is F-characteristic if any group containing an $$\mathfrak {X}$$-subgroup of finite index has also a characteristic subgroup of finite index that belongs to $$\mathfrak {X}$$. Thus the class $$\mathfrak {A}$$ of abelian groups is F-characteristic. The aim of this paper is to prove that many interesting classes of infinite groups are F-characteristic. Moreover, it is shown that the class of free groups and that of free abelian groups are not F-characteristic.

Large characteristic subgroups in which normality is a transitive relation

Large characteristic subgroups in which normality is a transitive relation

Large characteristic subgroups and abstract group classes

It is well-known that every virtually abelian group contains an abelian characteristic subgroup of finite index. We shall say that a group class is F-characteristic if any group containing an -subgroup of finite index has also a characteristic subgroup of finite index that belongs to . Thus the class of abelian groups is F -characteristic. Many other relevant group classes have recently been proved to be F-characteristic, and the aim of this paper is to give a further contribution to this topic, in particular in the context of classes of generalized nilpotent groups.

Large characteristic subgroups with modular subgroup lattice

It is known that if a group contains an abelian subgroup of finite index, then it also has an abelian characteristic subgroup of finite index. The aim of this paper is to prove that corresponding results hold when abelian subgroups are replaced either by subgroups having a modular subgroup lattice or by quasihamiltonian subgroups.

A Note on Large Characteristic Subgroups

ABSTRACTIt is known that if a group G has an abelian subgroup of finite index n, then it contains an abelian characteristic subgroup A of index at most nn. The aim of this paper is to improve this bound by showing that the characteristic subgroup A can be chosen of index at most n2. Examples prove that this bound is the best possible. Our main result is obtained as an application of a general method for the construction of large characteristic subgroups.

Large and symmetric: The Khukhro–Makarenko theorem on laws — without laws

We prove a generalisation of the Khukhro–Makarenko theorem on large characteristic subgroups with laws. This general fact implies new results on groups, algebras, and even graphs and other structures. Concerning groups, we obtain, e.g., a fact in a sense dual to the Khukhro–Makarenko theorem. A graph-theoretic corollary is an analogue of this theorem in which planarity plays the role of a multilinear identity. Also, we answer a question of Makarenko and Shumyatsky.

A short proof of the Khukhro-Makarenko theorem on large characteristic subgroups with laws

We give a short proof and a somewhat stronger version of the Khukhro-Makarenko theorem that each group which virtually satisfies an outer commutator identity contains a finite-index characteristic subgroup satisfying this identity. An estimate for the index of this characteristic subgroup is obtained.Bibliography: 3 titles.

Large characteristic subgroups satisfying multilinear commutator identities

The transformation of normal subgroups of finite index or Corank, satisfying a multilinear commutator identity into characteristic subgroups is presented. The reduction to the nilpotent groups is carried out by using Hall-Higman-type theorems. A characteristic subgroup with a finite index bounded in terms of n, and the weight of the commuter is obtained with the original subgroup having a finite index n. Similar results are proved for a subgroup that satisfies the identity of an arbitrary multilinear commutator, as well as for a nilpotent normal subgroup with a bound for the rank of the quotient group.

Large characteristic subgroups satisfying multilinear commutator identities

Large characteristic subgroups satisfying multilinear commutator identities