Hirsch Plotkin Radical Research Articles

On groups in which subnormal subgroups of infinite rank are commensurable‎ ‎with some normal subgroup

We study soluble groups G in which each subnormal subgroup H with infinite rank is commensurable with a normal subgroup, i.e. there exists a normal subgroup N such that the intersection of H and N has finite index in both H and N. We show that if such a G is periodic, then all subnormal subgroups are commensurable with a normal subgroup, provided either the Hirsch-Plotkin radical of G has infinite rank or G is nilpotent-by-abelian (and has infinite rank).

On metanilpotent groups satisfying the minimal condition on normal subgroups

Abstract A comprehensive account is given of the theory of metanilpotent groups with the minimal condition on normal subgroups. After reviewing classical material, many new results are established relating to the Fitting subgroup, the Hirsch–Plotkin radical, the Frattini subgroup, splitting and conjugacy, the Schur multiplier, Sylow structure and the maximal subgroups. Module theoretic and homological methods are used throughout.

Engel groups with an identity

We give an affirmative answer to the question whether a residually finite Engel group satisfying an identity is locally nilpotent. More generally, for a residually finite group [Formula: see text] with an identity, we prove that the set of right Engel elements of [Formula: see text] is contained in the Hirsch–Plotkin radical of [Formula: see text]. Given an arbitrary word [Formula: see text], we also show that the class of all groups [Formula: see text] in which the [Formula: see text]-values are right [Formula: see text]-Engel and [Formula: see text] is locally nilpotent is a variety.

Groups with bounded centralizer chains and the Borovik–Khukhro conjecture

Abstract Let G be a locally finite group and let F ⁢ ( G ) {F(G)} be the Hirsch–Plotkin radical of G. Let S denote the full inverse image of the generalized Fitting subgroup of G / F ⁢ ( G ) {G/F(G)} in G. Assume that there is a number k such that the length of every nested chain of centralizers in G does not exceed k. The Borovik–Khukhro conjecture states, in particular, that under this assumption, the quotient G / S {G/S} contains an abelian subgroup of finite index bounded in terms of k. We disprove this statement and prove a weak analogue of it.

On Local Nilpotency of the Normal Subgroups of a Group

A group G is said to have property μ whenever N is a non-locally nilpotent normal subgroup of G, there is a finite non-nilpotent G-quotient of N. FC-groups and groups with property ν satisfy property μ, where a group G is said to have property ν if every non-nilpotent normal subgroup of G has a finite non-nilpotent G-quotient. HP(G) is the Hirsch-Plotkin radical of G, and Φf(G) is the intersection of all the maximal subgroups of finite index in G (here Φf(G)=G if no such maximal subgroups exist). It is shown that a group G has property μ if and only if HP(G/Φf(G))=HP(G)/Φf(G) and that the class of groups with property ν is a proper subclass of that of groups with property μ. Also, the structure of the normal subgroups of a group: nilpotency with the minimal condition, is investigated.

Stability groups of series in vector spaces

We study the relationships between the subsets of left and bounded left Engel elements and various canonical locally nilpotent subgroups, in particular the Hirsch–Plotkin radical and the Fitting subgroup, of the stability groups of series of subspaces in vector spaces over fields and division rings. In this we extend work of C. Casolo and O. Puglisi and of G. Traustason.

Groups of infinite rank with finite conjugacy classes of subnormal subgroups

A group is called a V-group if it has finite conjugacy classes of subnormal subgroups. It is proved here that if G is a periodic soluble group in which every subnormal subgroup of infinite rank has finitely many conjugates, then G is a V-group, provided that its Hirsch–Plotkin radical has infinite rank. Corresponding results for periodic soluble groups in which every subnormal subgroup of infinite rank has finite index in its normal closure and for those in which every subnormal subgroup of infinite rank is finite over its core, are also obtained. Moreover, it is shown that the assumption on the Hirsch–Plotkin radical can be avoided in the case of periodic groups with nilpotent commutator subgroup.

On the Hirsch–Plotkin radical of stability groups

We study the stability group of subspace series of infinite dimensional vector spaces. In [1], the authors proved that when the vector space has countable dimension then the Hirsch–Plotkin radical of the stability group coincides with the set of all space automorphisms that fix a finite subseries and they conjectured that this would hold in all dimensions. We give a counter example of dimension 2ℵ0. We however show that in general the result remains true if the Hirsch–Plotkin radical is replaced by the Fitting group, the product of all the normal nilpotent subgroups of the stability group. We also show that the Hirsch–Plotkin radical has a certain strong local nilpotence property.

Left 3-Engel elements in groups of exponent 5

It is still an open question whether a left 3-Engel element of a group G is always contained in the Hirsch–Plotkin radical of G. In this paper we begin a systematic study of this problem. The problem is first rephrased as saying that a certain type of groups are locally nilpotent. We refer to these groups as sandwich groups as they can be seen as the analogs of sandwich algebras in the context of Lie algebras. We show that any 3-generator sandwich group is nilpotent and obtain a power-conjugation presentation for the free 3-generator sandwich group. As an application we show that the left 3-Engel elements in any group G of exponent 5 are in the Hirsch–Plotkin radical of G.

Hirsch–Plotkin radical of stability groups

We study the Hirsch–Plotkin radical of stability groups of (general) subspace series of infinite dimensional vector spaces. We show that in countable dimension and some other cases, the HP-radical of the stability group coincides with the set of all space automorphisms that fix a finite subseries; this implies that the Hirsch–Plotkin radical is a Fitting group. Conversely, we prove that every countable Fitting group, which is either torsion-free or a p-group may be represented as a subgroup of the Hirsch–Plotkin radical of a series stabilizer.

On Certain Characterizations of Barely Transitive Groups

In the present work we study perfect barely transitive groups in which a point stabilizer has non-trivial Hirsch–Plotkin radical and give characterizations of these groups. We also prove that a perfect barely transitive group has two non-trivial proper subgroups which intersect trivially.

ON SOLUBILITY OF GROUPS WITH BOUNDED CENTRALIZER CHAINS

AbstractThe c-dimension of a group is the maximum length of a chain of nested centralizers. It is proved that a periodic locally soluble group of finite c-dimension k is soluble of derived length bounded in terms of k, and the rank of its quotient by the Hirsch–Plotkin radical is bounded in terms of k. Corollary: a pseudo-(finite soluble) group of finite c-dimension k is soluble of derived length bounded in terms of k.

Extensions of Periodic Linear Groups with Finite Unipotent Radical

A group is called p-linear if it is isomorphic to a subgroup of GL(n,K) for some field K of characteristic p and some integer n. Let H be a normal subgroup of G and assume that both H and G/H are periodic and p-linear. In addition, assume that both H and G/H have finite unipotent radicals and that the Hirsch-Plotkin radical of H is C˘ernikov. The main result of this article is a proof that under these assumptions G is p-linear. An example is provided showing the result is false if the assumption regarding the Hirsch-Plotkin radical is removed.

On the Hirsch-Plotkin radical of a factorized group

Let the group G = AB be the product of two subgroups A and B. A normal subgroup K of G is said to be factorized if K = (A ∩ K)(B ∩ K) and A ∩ B ≤ K, and this is well-known to be equivalent to the fact that K = AK ∩ BK (see [1]). Easy examples show that normal subgroups of a product of two groups need not, in general, be factorized. Therefore the determination of certain special factorized subgroups is of relevant interest in the investigation concerning the structure of a factorized group. In this direction E. Pennington [5] proved that the Fitting subgroup of a finite product of two nilpotent groups is factorized. This result was extended to infinite groups by B. Amberg and theauthors, who provedin [2] that if the soluble group G = AB with finite abelian section rank isthe product of two locally nilpotent subgroups A and B, then the Hirsch-Plotkin radical (i.e. the maximum locally nilpotent normal subgroup) of G is factorized. If G is a soluble ℒI group and the factors A and B are nilpotent, it was shown in [3] that also the Fitting subgroup of G is factorized. However, Pennington's theorem becomes false for finite soluble groups which are the productof two arbitrary subgroups. For instance, the symmetric group of degree 4 is the product of a subgroup isomorphic with the symmetric group of degree 3 and a cyclic subgroup of order 4, but its Fitting subgroup is not factorized.

Functorial radicals and non-abelian torsion

Radicals appear in many algebraic contents. For modules over a ring, they give rise to pre-torsion and torsion theories, Goldman (5), Lambek (14). In the category of groups, Kurosh, Plotkin and others have introduced radicals (6), (13), (21), but unlike the radicals in module theory these radicals are not necessarily functorial, as for example the nil radical and the Hirsch-Plotkin radical (6). The functorial method in module theory has been extended to abelian categories, Dickson (2), to the category of nilpotent groups, Hilton (8), Warfield (25), and to the category of groups, Plotkin (22), and to general categories, Wiegandt (26), Holcombe and Walker (10).

Locally soluble groups with min-n.

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 countable groups G which have hypercentral nonnilpotent Hirsch-Plotkin radical, and which at the same time are isomorphic to all their non-trivial homomorphic images.