Finite Soluble Groups Research Articles

A note on conjugacy of supplements in finite soluble groups

Let G be a finite soluble group and Q be a nilpotent normal subgroup of G. Here we prove that if no G-chief factor of G/Q is G-isomorphic to a G-chief factor of Q, then certain supplements to Q in G are G-conjugate.

Counterexamples to a fiber theorem

We exhibit a counterexample to a fiber theorem stated by F. Fumagalli in [Francesco Fumagalli, On the homotopy type of the Quillen complex of finite soluble groups, J. Algebra 283 (2) (2005) 639–654] and show how it affects the rest of Fumagalli's paper. As a consequence, whether the poset A p ( G ) is homotopy equivalent to a wedge of spheres for any finite solvable group G seems to remain an open question.

Finite Soluble Group Generated by the Conjugacy Class of an Involution

Let G be a finite group and P a subgroup of order 2. We study in this article the structures of the soluble subgroup of G that is generated by three conjugates of P. We use the results we proved about the soluble subgroups that are generated by three conjugates of P to find a soluble analogue of the Baer–Suzuki Theorem in the case prime 2.

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.

Infinite groups with many permutable subgroups

A subgroup H of a group G is said to be permutable in G , if HK = KH for every subgroup K of G . A result due to Stonehewer asserts that every permutable subgroup is ascendant although the converse is false. In this paper we study some infinite groups whose ascendant subgroups are permutable ( AP -groups). We show that the structure of radical hyperfinite AP -groups behave as that of finite soluble groups in which the relation to be a permutable subgroup is transitive ( PT -groups).

Persistent characterizations of injectors in finite solvable groups

In response to a question of Doerk and Hawkes [Finite soluble groups, de Gruyter, 1992, p. 553], we shall obtain characterizations of the injectors of a finite solvable group (without recourse to the concept of a Fitting set), and we also answer in the negative a question in [Dark and Feldman, J. Group Theory 9: 2006, p. 785].

On the prefrattini subgroups of finite soluble groups

We study the partially prefrattini groups of a finite soluble group. We prove that the set of all partially prefrattini subgroups associated with the Gaschutz system of complements to crowns is a Boolean lattice.

On Injectors Of Finite Soluble Groups

In this article, we give the description of the ℌ-injectors of a finite soluble group, for a Hartley class ℌ.

Soluble products of connected subgroups

The main result in the paper states the following: For a finite group G=AB , which is the product of the soluble subgroups A and B , if \langle a,b \rangle is a metanilpotent group for all a\in A and b\in B , then the factor groups \langle a,b \rangle F(G)/F(G) are nilpotent, F(G) denoting the Fitting subgroup of G . A particular generalization of this result and some consequences are also obtained. For instance, such a group G is proved to be soluble of nilpotent length at most l+1 , assuming that the factors A and B have nilpotent length at most l . Also for any finite soluble group G and k\geq 1 , an element g\in G is contained in the preimage of the hypercenter of G/F_{k-1}(G) , where F_{k-1}(G) denotes the ( k-1 )th term of the Fitting series of G , if and only if the subgroups \langle g,h\rangle have nilpotent length at most k for all h\in G .

Finite groups with minimal 1-PIM

Let \(\mathbb F\) be a field of characteristic \(\ell > 0\) and let G be a finite group. It is well-known that the dimension of the minimal projective cover \(\Phi_1^G\) (the so-called 1-PIM) of the trivial left \(\mathbb F[G]\) -module is a multiple of the \(\ell\) -part \(|G|_\ell\) of the order of G. In this note we study finite groups G satisfying \(\dim_{\mathbb F}(\Phi_1^G)=|G|_\ell\) . In particular, we classify the non-abelian finite simple groups G and primes \(\ell\) satisfying this identity (Theorem A). As a consequence we show that finite soluble groups are precisely those finite groups which satisfy this identity for all prime numbers \(\ell\) (Corollary B). Another consequence is the fact that the validity of this identity for a finite group G and for a small prime number \(\ell\in\{2,3,5\}\) implies the existence of an \(\ell^\prime\) -Hall subgroup for G (Theorem C). An important tool in our proofs is the super-multiplicativity of the dimension of the 1-PIM over short exact sequences (Proposition 2.2).

On 2-generated subgroups and products of groups

For a non-empty class of groups ℱ, two subgroups A and B of a finite group G are said to be ℱ-connected if 〈a, b〉 ∈ ℱ for all a ∈ A and b ∈ B. This paper is a study of ℱ-connection for saturated formations ℱ ⊆ (where denotes the class of all finite groups with nilpotent commutator subgroup). The class of all finite supersoluble groups constitutes an example of such a saturated formation. It is shown for example that in a finite soluble group G = AB the subgroups A and B are -connected if and only if [A, B] ⩽ F(G), where F(G) denotes the Fitting subgroup of G. Also ℱ-connected finite soluble products for any saturated formation ℱ with ℱ ⊆ are characterized.

On the Fitting height of a soluble group that is generated by a conjugacy class of 3-elements

Let G be a finite soluble group that is generated by a conjugacy class consisting of elements of order 3. We show that there exist four conjugates of an element of order 3 that generate a subgroup with the same Fitting height as G. We use this result to find a soluble analogue of the Baer–Suzuki theorem in the case prime 3.

On Certain Distributive Lattices of Subgroups of Finite Soluble Groups

In this paper, we prove the following result. Let \({\mathfrak{F}}\) be a saturated formation and Σ a Hall system of a soluble group G. Let X be a w-solid set of maximal subgroups of G such that Σ reduces into each element of X. Consider in G the following three subgroups: the \({\mathfrak{F}}\)-normalizer D of G associated with Σ; the X-prefrattini subgroup W = W(G,X) of G; and a hypercentrally embedded subgroup T of G. Then the lattice \({\mathfrak{L}(T,W,D)}\) generated by T,D and W is a distributive lattice of pairwise permutable subgroups of G with the cover and avoidance property.

A characterization of finite soluble groups

A characterization of finite soluble groups

FINITE AXIOMATIZATION OF FINITE SOLUBLE GROUPS

It is proved that the finite soluble groups can be characterized among finite groups by a first-order sentence, namely, the sentence that asserts that no non-trivial element g is a product of 56 commutators [x, y] with entries x, y conjugate to g.

The Concept of Primitivity in Group Theory and the Second Memoir of Galois

The so-called Second Memoire of Evariste Galois, written in 1 830 and first published in 1 846, is notoriously difficult to understand. Already the title 'Des equations primitives qui sont solubles par radicaux' requires considerable thought. For, the word 'primitive', which has a standard meaning in the context of group theory now, had no such context or meaning when Galois used it. I have two goals in the present paper: first, to give an account of the development of the concept of primitivity in finite group theory in the 19th century; secondly, to provide commentary on the first part of the Second Memoire. These may appear to be two separate projects for which two separate papers would be appropriate. I combine them into one because, as the reader will see, they are in fact so closely related as to be almost inseparable. Hence my choice of an ambiguous title. I also have two categories of reader in mind: historians and mathematicians. For the former, but also because one cannot discuss the history of the mathematics without having the details in front of one, I give expositions of some basic theory of equations and groups. For the latter I rehearse some of the known facts about Galois' writings. Almost certainly the Second Memoire was written in 1830 (see [18, p. 494]). It was first published in 1846 by Liouville [15] and there have been several re-publications since, culminating in the critical edition of 1962 by Robert Bourgne and J.-P. Azra [18]. It is an unfinished first draft, nowhere near as much revised or as important as the Premier Memoire. Nevertheless, where the Premier Memoire describes what we now think of as Galois Theory, the Second Memoire focusses heavily on groups and, as we shall see, has had, through the work of Camille Jordan and others, an important influence on group theory. It provides moreover a significant piece of evidence about the workings of the mind of an extraordinarily creative and intuitive young genius. Either one of these facts would be reason to make it worthy of intensive study; together they are compelling. The Second Memoire is in two parts. The first is about finite soluble equations and groups, the second about the groups we now call AGL (2, p) and PSL (2, p). The present paper is concerned only with the first part I propose to write about the second part on some other occasion.

The G-strong containment for locally defined formations of finite soluble groups

Let F and H be two saturated formations of finite soluble groups, with canonical definition F and H, respectively, and Char ( F ) ⊆ Char ( H ) . We say that F is G -strong containment in H , when for all p ∈ Char ( F ) and for all H ∈ H the H ( p ) -Residual of H is in a F -projector E of H containment. In this case we write F ≪ G H . In this work we wants to show a characterization of the saturated formation H , which contain G -strongly the formations S π , NX , N k and U ¨ , respectively.

The Structure of G/ Φ (G) in Terms of σ (G)

Let G be a finite group. We let [Formula: see text] and σ (G) denote the number of maximal subgroups of G and the least positive integer n such that G is written as the union of n proper subgroups, respectively. In this paper, we determine the structure of G/ Φ (G) when G is a finite soluble group with [Formula: see text].

Characterization of injectors in finite soluble groups

Characterization of injectors in finite soluble groups

A Note on the σ-Covers of Finite Soluble Groups

A cover for a finite group is a set of proper subgroups whose union is the whole group. For a finite group G, we denote by σ (G) the least positive integer n such that G has a cover of size n. This paper deals with the covers of a finite soluble group G having σ (G) elements of maximal subgroups.