Finite Soluble Groups Research Articles

Fixed Points in Finite Soluble Groups#

Let A be an elementary abelian group of order at least p 3 acting on a finite p′-group G that is soluble with derived length d. Assume that γ c (C G (a)) has exponent dividing m for any a ∈ A #. It is proved that there exist {p, d, c, m}-bounded numbers c 1 and m 1 such that γ c 1 (G) has exponent dividing m 1.

SATURATED FORMATIONS CLOSED UNDER SYLOW NORMALIZERS

In this article we show that a finite soluble group possesses nilpotent Hall subgroups for well-defined sets of primes if and only if its Sylow normalizers satisfy the same property. In fact, this property of groups provides a characterization of the subgroup-closed saturated formations, whose elements are characterized by the Sylow normalizers belonging to the class, in the universe of all finite soluble groups.

ON INDICES OF SUBNORMAL SUBGROUPS OF FINITE SOLUBLE GROUPS#

All groups considered in this paper are finite soluble groups. Let t p (G) = max H◃◃G {j|p j ||| H G : H |} and t (G) = max p ∈π( G ) t p ( G ). In this paper, we study the influence of t ( G ) on the structure of a finite soluble group G . In particular, the bounds of the nilpotent length (derived length, p -length) of a soluble group G with t ( G ) ≤ 2 are found.

On two conditions on characters and conjugacy classes in finite soluble groups

On two conditions on characters and conjugacy classes in finite soluble groups

Soluble Groups with an Automorphism Inverting Many Elements

Let G be a finite soluble group. We derive upper bounds, in terms of the derived length of G, for the maximal proportion of elements of G that can be sent to their inverses under a group automorphism.

On Products of Nonnormal Subgroups of Finite Soluble Groups

We study subgroup-closed saturated formations and Fitting formations closed by taking the products of nonnormal subgroups of finite soluble groups. A constructive description of soluble subgroup-closed saturated formations and Fitting formations of finite groups with a given property is obtained.

A Characterisation of Injectors of Finite Soluble Groups

A Characterisation of Injectors of Finite Soluble Groups

A Characteristic Property of Finite p -Nilpotent Groups#

An element x of a group G is called Q-central if there exists a central chief factor H/K of G such that x ∈ H\\K. It is proved that a finite group G is p-nilpotent if and only if every element in G p \\Φ(G p ) is Q-central. We will adapt Doerk and Hawkes [Doerk, K., Hawkes, T. (1992). Finite Soluble Groups. Berlin–New York: Walter de Gruyter, pp. 892] for notations and basic results.

Soluble Groups of Small Wielandt Length#

In this paper we are interested in the relationship between the Wielandt series and strong Wielandt series of a finite soluble group. Some exact bounds on strong Wielandt length in terms of Wielandt length are obtained for groups with p-length 1 for all primes p. We also give two new results relating the Wielandt subgroup and strong Wielandt subgroup of an arbitrary finite soluble group.

On the homotopy type of the Quillen complex of finite soluble groups

We prove that the homotopy type of the Quillen complex of a finite soluble group at the prime p ≠ 2 is that of a wedge of spheres of possible different dimensions.

Special polycyclic generating sequences for finite soluble groups

Polycyclic generating sequences are known to be a powerful tool in the design of practical and efficient algorithms for computing in finite soluble groups. Here we describe a further development: the so-called special polycyclic generating sequences. We give an overview of their properties and introduce a practical algorithm for determining a special polycyclic generating sequence in a given finite soluble group.

Indices of Maximal Subgroups of Finite Soluble Groups

We look at the structure of a soluble group G depending on the value of a function m(G)= max mpG), where mp(G)=max{logp|G:M| | M k. Moreover, every two subgroups of G enjoying (1) and (2) are mutually conjugate.

G-Covering Systems of Subgroups for Classes of p-Supersoluble and p-Nilpotent Finite Groups

Let \(F\) be a class of groups. Given a group G, assign to G some set of its subgroups Σ=Σ(G). We say that Σ is a G-covering system of subgroups for \(F\) (or, in other words, an \(F\)-covering system of subgroups in G) if G∈\(F\) whenever either Σ=O or Σ≠O and every subgroup in Σ belongs to \(F\). We find the systems of subgroups in the class of finite soluble groups G which are simultaneously the G-covering systems of subgroups for the classes of p-supersoluble and p-nilpotent groups.

Fitting classes and lattice formations I

AbstractA lattice formation is a class of groups whose elements are the direct product of Hall subgroups corresponding to pairwise disjoint sets of primes. In this paper Fitting classes with stronger closure properties involving F-subnormal subgroups, for a lattice formation F of full characteristic, are studied. For a subgroup-closed saturated formation G, a characterisation of the G-projectors of finite soluble groups is also obtained. It is inspired by the characterisation of the Carter subgroups as the N-projectors, N being the class of nilpotent groups.

On Local Embedding Properties of Injectors of Finite Soluble Groups

AbstractIn the present paper we consider Fitting classes of finite soluble groups which locally satisfy additional conditions related to the behaviour of their injectors. More precisely, we study Fitting classes 1 ≠⊆such that an-injector of G is, respectively, a normal, (sub)modular, normally embedded, system permutable subgroup of G for all G ∈.Locally normal Fitting classes were studied before by various authors. Here we prove that some important results—already known for normality—are valid for all of the above mentioned embedding properties. For instance, all these embedding properties behave nicely with respect to the Lockett section. Further, for all of these properties the class of all finite soluble groups G such that an x-injector of G has the corresponding embedding property is not closed under forming normal products, and thus can fail to be a Fitting class.

Saturated formations and Sylow normalisers

Sufficient conditions are provided in order that some classes of finite soluble groups, defined by properties of the Sylow normalisers, are saturated formations.

A note on subgroup-closed Fitting classes of finite soluble groups

A note on subgroup-closed Fitting classes of finite soluble groups

On locally normal Fitting classes of finite soluble groups

Let X , F , X ⊆ F , be non-trivial Fitting classes of finite soluble groups such that G X is an X -injector of G for all G∈ F . Then X is said to be normal in F ( F -normal). We show that for a subgroup-closed Fitting class X the collection of all subgroup-closed Fitting classes in which X is normal forms a complete, distributive and atomic lattice. Moreover, X is determined uniquely by the unique maximal subgroup-closed Fitting class in which X is normal, and in many cases there is an algorithm to describe this class explicitly. Further we show that if F is a subgroup-closed Fitting class such that a unique minimal subgroup-closed F -normal Fitting class exists, the collection of all subgroup-closed F -normal Fitting classes also forms a complete distributive lattice which in addition is dual atomic if F is of bounded nilpotent length.

Finite soluble groups whose subnormal subgroups permute with certain classes of subgroups

Finite soluble groups whose subnormal subgroups permute with certain classes of subgroups

On finite soluble groups in which Sylow permutability is a transitive relation

A characterisation of finite soluble groups in which Sylow permutability is a transitive relation by means of subgroup embedding properties enjoyed by all the subgroups is proved. The key point is an extension of a subnormality criterion due to Wielandt.