Subgroups Of Finite Soluble Groups Research Articles

On σ-Residuals of Subgroups of Finite Soluble Groups

Let σ={σi:i∈I} be a partition of the set of all prime numbers. A subgroup H of a finite group G is said to be σ-subnormal in G if H can be joined to G by a chain of subgroups H=H0⊆H1⊆⋯⊆Hn=G where, for every j=1,⋯,n, Hj−1 is normal in Hj or Hj/CoreHj(Hj−1) is a σi-group for some i∈I. Let B be a subgroup of a soluble group G normalising the Nσ-residual of every non-σ-subnormal subgroup of G, where Nσ is the saturated formation of all σ-nilpotent groups. We show that B normalises the Nσ-residual of every subgroup of G if G does not have a section that is σ-residually critical.

On two questions from the Kourovka Notebook

The aim of this paper is to give answers to some questions concerning intersections of system normalisers and prefrattini subgroups of finite soluble groups raised by the third author, Shemetkov and Vasil'ev in the Kourovka Notebook [10]. Our approach depends on results on regular orbits and it can be also used to extend a result of Mann [9] concerning intersections of injectors associated to Fitting classes.

Maximal subgroups of finite soluble groups in general position

For a finite group G we investigate the difference between the maximum size $${{\mathrm{MaxDim}}}(G)$$ of an “independent” family of maximal subgroups of G and maximum size m(G) of an irredundant sequence of generators of G. We prove that $${{\mathrm{MaxDim}}}(G)=m(G)$$ if the derived subgroup of G is nilpotent. However, $${{\mathrm{MaxDim}}}(G)-m(G)$$ can be arbitrarily large: for any odd prime p, we construct a finite soluble group with Fitting length two satisfying $$m(G)=3$$ and $${{\mathrm{MaxDim}}}(G)=p$$ .

Pronormality and Strong Pronormality of Subgroups

A subgroup H of a group G is said to be pronormal if, for any element g ∈ G, subgroups H and H g are conjugate in $$ \left\langle {H,{H^g}} \right\rangle $$ . A subgroup H of a group G is said to be strongly pronormal if, for any subgroup K ≤ H and any element g ∈ G, there exists an element $$ x\in \left\langle {H,{K^g}} \right\rangle $$ such that K gx ≤ H. Many known examples of pronormal subgroups, namely, normal subgroups, maximal subgroups, Sylow subgroups of finite groups, and Hall subgroups of finite soluble groups, will also exemplify strongly pronormal subgroups. It is shown that Carter subgroups of finite groups (which are always pronormal) are not strongly pronormal in general, even in soluble groups.

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.

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.

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.

Subgroups of finite soluble groups inducing the same permutation character

In this paper there are found necessary and sufficient conditions that a pair of solvable finite groups, say G G and K K , must satisfy for the existence of a solvable finite group L L containing two isomorphic copies of G G and H H inducing the same permutation character. Also a construction of L L is given as an iterated wreath product, with respect to their actions on their natural modules, of finite one-dimensional affine groups.

Permutably embedded subgroups of finite soluble groups

Permutably embedded subgroups of finite soluble groups

On [formula omitted]-Normal Subgroups of Finite Soluble Groups

On [formula omitted]-Normal Subgroups of Finite Soluble Groups

Nilpotent subgroups of finite soluble groups

Nilpotent subgroups of finite soluble groups

Intersecting subgroups of finite soluble groups

This paper describes an algorithm for constructing certain important subgroup intersections H ∩ K, of a finite soluble group G. The algorithm is applicable if the index of H in G is coprime to the index of K in G, or more generally, if each hypereccentric factor of G is covered by H or K. If G has composition length n, then the intersection algorithm computes H ∩ K using O( n 3 log| G|) group multiplications. This intersection algorithm can be applied to compute Sylow bases, system normalisers and complements to abelian terms of the lower nilpotent series. These structures are particularly useful for constructing the automorphism group of G. Intersecting arbitrary subgroups of a finite soluble group is, in general, a difficult problem. This is shown by an example where the subgroups H and K of G both avoid an eccentric factor of G, and computing H ∩ K is equivalent to constructing a discrete logarithm in a finite field. However, there are no known polynomial algorithms for constructing discrete logarithms.

Pronormal subgroups and homomorphs in finite groups

A local version of the theory of homomorphs and Schunck classes is given. It is shown that for any finite soluble group the pronormal subgroups are precisely the covering subgroups with respect to “Schunck sets” in this group. As an application simple proofs of some results on pronormal subgroups of finite soluble groups are obtained. Finally a question of Doerk is answered in the negative: any finite soluble group is a subgroup of a minimal non-trivial pronormal subgroup of some finite soluble group.

A class of modules over a locally finite group III

Let G be a locally finite group, k a field of characteristic p ≥ 0 and V a right kG-module. We say that V is an -module over kG, if each p′-subgroup H of G contains a finite subgroup F with the same fixed points as H in V. (By convention, 0′ is taken as the set of all primes.) Such modules arise as elementary abelian section of -groups, a class of locally finite groups similar in many ways to the class of finite soluble groups.The main theorem is that if V is an -module over kG with trivial Frattini submodule, and G is almost abelian, then every composition factor of V is complemented. This is a crucial ingredient in Tomkinson's theory of prefrattini subgroups in a certain subclass of . An example is given to show that the theorem breaks down for metabelian G. This leads to an example of a -group in which there are no analogues of prefrattini subgroups - the first situation where one of the standard conjugacy classes of subgroups of finite soluble groups has no decent analogue in the whole class

On subgroups of finite soluble groups III

We define a notion of generalized normality for subgroups of finite soluble groups. We show that a normalizer relative to this notion exists and is homomorphism-invariant. We make comparisons with previous constructions, and develop briefly a general theory of normality relations and normalizers.

On Certain Sublattices of the Lattice of Subgroups Generated by the Prefrattini Subgroups, the Injectors and the Formation Subgroups

Various characteristic conjugacy classes of subgroups having covering/avoidance properties with respect to chief factors have recently played a major role in the study of finite soluble groups. Apart from the subgroups which are now called Hall subgroups, P. Hall [7] also considered the system normalizers of a finite soluble group and showed that these form a characteristic conjugacy class, cover the central chief factors and avoid the rest. The system normalizers were later shown by Carter and Hawkes [1] to be the simplest example of a wealth of characteristic conjugacy classes of subgroups of finite soluble groups which arise naturally as a consequence of the theory of formations.

On subgroups of finite solvable groups II

We define a notion of generalized normality for subgroups of finite soluble groups. We show that a normalizer relative to this notion exists and is homomorphism-invariant. We make comparisons with previous constructions, and develop briefly a general theory of normality relations and normalizers.

P-Normally embedded subgroups of finite soluble groups

p-Normally embedded subgroups of finite soluble groups

Another characteristic conjugacy class of subgroups of finite soluble groups

Let name be a class of finite soluble groups with the properties: (1) is a Fitting class (i.e. normal subgroup closed and normal product closed) and (2) if N ≦ H ≦ G ∈, N ⊲ G and H/N is a p-group for some prime p, then H ∈. Then is called a Fischer class. In any finite soluble group G, there exists a unique conjugacy class of maximal -subgroups V called the -injectors which have the property that for every N◃◃G, N ∩ V is a maximal -subgroup of N [3]. 3. By Lemma 1 (4) [7] an -injector V of G covers or avoids a chief factor of G. As in [7] we will call a chief factor -covered or -avoided according as V covers or avoids it and -complemented if it is complemented and each of its complements contains some -injector. Furthermore we will call a chief factor partially-complemented if it is complemented and at least one of its complements contains some -injector of G.

Nilpotent subgroups of finite soluble groups

The general problem, with a particular instance of which the present paper is concerned, is to obtain a description of the local structure of a group from information about the global structure. The aspect of local structure investigated here is the embedding of subgroups, especially of nilpotent subgroups in finite soluble groups. A classification of embeddings of subgroups in finite groups by means of an arithmetic function called abnormal depth was proposed in [6]. Let H be a subgroup of a finite group G. Then a(G:H), the abnormal depth of H in G, is the least number of abnormal links appearing in any balanced chain of subgroups connecting H to G, that is a chain for which each link is either normal or abnormal. Thus a (G:H)= 0 if and only if H is subnormal in G; and a(G:P)__< 1 for every subgroup P of G of prime power order. It was shown in [6] that if H is a nilpotent subgroup of a finite soluble group G, of nilpotent length n, then a (G: H) =< n - 1. Here in w 1 we examine in greater detail the easiest non-trivial case, in which n = 2, and then in w 2 prove certain supplementary results for n = 3 and n = 4. Some simple wreath product properties are established in w 3 and used in w 4 for the construction of examples showing that the embedding results obtained cannot be improved in various obvious ways. Notation and terminology follow common usage. If t; and ~ are classes of groups, then 3s ~ denotes the class of all groups G having a normal subgroup X such that X e 3~ and G/X e ~. This defines a composition of classes of groups which in general is not associative. However, we shall deal only with classes of which the composition is associatNe, and we may therefore omit brackets from products of more than two classes. Since we shall be concerned exclusively with finite groups, we take 91 to denote the class of finite nilpotent groups and 9.1 the class of finite abelian groups. Then for any positive integer n, 9l" is the class of finite soluble groups of nilpotent lengths <__ n; and 9.I" is the class of finite soluble groups of derived lengths __< n. Henceforth the term group is understood to mean finite group. Then any group G has a unique smallest normal subgroup L such that G/L is nilpotent: G/L is called the 91-residual ofG. IfH is any subgroup of G, then there is a unique smallest normal subgroup of G containing H, called the normal closure of H in G and denoted by Ha; and a unique smallest subnormal subgroup of G containing H, called the subnormal closure of H in G and (following Wielandt [8]) denoted by H'" a. If H a = G, we shall say that H is contranormal in G. Then, for any subgroup H of G, it is clear that H is contranormal in H'" a. (This is to be compared with the fact that the hypernormalizer NE(H ) of H in G is self-normalizing in G.) An abnormal subgroup is both self-normalizing and