-normalizers and -covering subgroups

1969 ◽  
Vol 66 (2) ◽  
pp. 215-230 ◽  
Author(s):  
Mary Jane Prentice
Keyword(s):  
Normal Subgroup ◽  
Conjugacy Class ◽  
Soluble Group ◽  
Saturated Formation ◽  
Normal System ◽  
Finite Soluble Group ◽  
Chief Factors ◽  
Maximal Chains

In (1), Carter and Hawkes define the -normalizers of a finite soluble group for any saturated formation . These subgroups are conjugate, invariant under homomorphisms of the group, cover and avoid the chief factors of the group and may be characterized by means of the maximal chains of subgroups connecting them to the group. The first aim of the present paper is to generalize both the -normalizers and the relative system normalizers (Hall (5)) of a finite soluble group G. We choose an arbitrary normal subgroup X(p) of G for each prime p dividing the order of G, forming a normal system = {X(p)} of G. Each of these normal systems of G yields a conjugacy class of subgroups, called the -normalizers of G, which possess the above properties of -normalizers. However, they do not satisfy all the properties of the -normalizers unless is a so-called integrated normal system of G.

Prefrattini Subgroups and Cover-Avoidance Properties in 𝔘-Groups

1975 ◽  
Vol 27 (4) ◽  
pp. 837-851 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Conjugacy Class ◽  
Soluble Group ◽  
Saturated Formation ◽  
Conjugacy Classes ◽  
Pronormal Subgroup ◽  
Finite Soluble Group ◽  
Factors Associated ◽  
Chief Factors

W. Gaschutz [5] introduced a conjugacy class of subgroups of a finite soluble group called the prefrattini subgroups. These subgroups have the property that they avoid the complemented chief factors of G and cover the rest. Subsequently, these results were generalized by Hawkes [12], Makan [14; 15] and Chambers [2]. Hawkes [12] and Makan [14] obtained conjugacy classes of subgroups which avoid certain complemented chief factors associated with a saturated formation or a Fischer class. Makan [15] and Chambers [2] showed that if W, D and V are the prefrattini subgroup, 𝔍-normalizer and a strongly pronormal subgroup associated with a Sylow basis S, then any two of W, D and V permute and the products and intersections of these subgroups have an explicit cover-avoidance property.

On f-Prefrattini Subgroups

1972 ◽  
Vol 15 (3) ◽  
pp. 345-348 ◽  
Author(s):  
Graham A. Chambers
Keyword(s):  
Conjugacy Class ◽  
Soluble Group ◽  
Analogous Result ◽  
Saturated Formation ◽  
Finite Soluble Group ◽  
Chief Factors

The Prefrattini subgroups of a finite soluble group were introduced by Gaschutz [3]. These are a conjugacy class of subgroups which avoid complemented chief factors and cover Frattini chief factors. Gaschutz [3, Satz 7.1] showed that if G has p-length 1 for each prime p, and if U≤G avoids all complemented chief factors and covers all Frattini factors, then U is a Prefrattini subgroup of G. We begin by proving the analogous result for the f-Prefrattini subgroups introduced by Hawkes [5], If f is a saturated formation, then the f-Prefrattini subgroups of G are a conjugacy class of subgroups which avoid f-eccentric complemented chief factors of G and cover all other chief factors of G.

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

1973 ◽  
Vol 25 (4) ◽  
pp. 862-869 ◽  
Author(s):  
A. R. Makan
Keyword(s):  
Conjugacy Class ◽  
Soluble Group ◽  
Conjugacy Classes ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
Hall Subgroups ◽  
Chief Factors ◽  
Lattice Of 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.

Another characteristic conjugacy class of subgroups of finite soluble groups

1970 ◽  
Vol 11 (4) ◽  
pp. 395-400 ◽  
Author(s):  
A. Makan
Keyword(s):  
Normal Subgroup ◽  
Conjugacy Class ◽  
Soluble Group ◽  
Fitting Class ◽  
Chief Factor ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
Normal Product

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.

scholarly journals The formation generated by a finite group

1970 ◽  
Vol 2 (3) ◽  
pp. 347-357 ◽  
Author(s):  
R. M. Bryant ◽  
R. A. Bryce ◽  
B. Hartley
Keyword(s):  
Finite Group ◽  
Soluble Group ◽  
Saturated Formation ◽  
Finite Soluble Group ◽  
Arbitrary Finite Group ◽  
A Finite Group

We prove here that the (saturated) formation generated by a finite soluble group has only finitely many (saturated) subformations. This answers a question asked by Professor W. Gaschütz. Some partial results are also given in the case of a formation generated by an arbitrary finite group.

scholarly journals On complemented chief factors of finite soluble groups

1972 ◽  
Vol 7 (1) ◽  
pp. 101-104 ◽  
Author(s):  
D.W. Barnes
Keyword(s):  
Soluble Group ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
Chief Factors ◽  
One To One

Let G = H0 > H1 > … > Hr = 1 and G = K0 > K1 > … > Kr =1 be two chief series of the finite soluble group G. Suppose Mi complements Hi/Hi+1. Then Mi also complements precisely one factor Kj/Kj+1, of the second series, and this Kj/Kj+1 is G-isomorphic to Hi/Hi+1. It is shown that complements Mi can be chosen for the complemented factors Hi/Hi+1 of the first series in such a way that distinct Mi complement distinct factors of the second series, thus establishing a one-to-one correspondence between the complemented factors of the two series. It is also shown that there is a one-to-one correspondence between the factors of the two series (but not in general constructible in the above manner), such that corresponding factors are G-isomorphic and have the same number of complements.

A note on system normalizers of a finite soluble group

1966 ◽  
Vol 62 (3) ◽  
pp. 339-346 ◽  
Author(s):  
T. O. Hawkes
Keyword(s):  
Soluble Group ◽  
Partial Answer ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
Sylow Subgroups ◽  
Chief Factors ◽  
A Chain

Introduction. Hall ((3), (4)) introduced the concept of a Sylow system and its normalizer into the theory of finite soluble groups. In (4) he showed that system normalizers may be characterized as those subgroups D of G minimal subject to the existence of a chain of subgroups from D up to G in which each subgroup is maximal and non-normal in the next; he also showed that a system normalizer covers all the central chief factors and avoids all the eccentric chief factors of G (for definitions of covering and avoidance, and an account of their elementary properties), the reader is referred to Taunt ((5)). This note arises out of an investigation into the question to what extent this covering/ avoidance property characterizes system normalizers; it provides a partial answer by means of two elementary counter-examples given in section 3 which seem to indicate that the property ceases to characterize system normalizers as soon as the ‘non-commutativity’ of the group is increased beyond a certain threshold. For the sake of completeness we include proofs in Theorems 1 and 2 of generalizations of two known results communicated to me by Dr Taunt and which as far as we know have not been published elsewhere. Theorem 1 shows the covering/avoidance property to be characteristic for the class of soluble groups with self-normalizing system normalizers introduced by Carter in (1), while Theorem 2 shows the same is true for A -groups (soluble groups with Abelian Sylow subgroups investigated by Taunt in (5)).

An example in the theory of solubl groups

1970 ◽  
Vol 67 (1) ◽  
pp. 13-16 ◽  
Author(s):  
T. O. Hawkes
Keyword(s):  
Soluble Group ◽  
Nilpotent Groups ◽  
Saturated Formation ◽  
Finite Soluble Group ◽  
Carter Subgroup ◽  
Soluble Groups ◽  
University Of Wisconsin ◽  
Unpublished Work ◽  
The University

Let G be a finite soluble group. In (1) Alperin proves that two system normalizers of G contained in the same Carter subgroup C of G are conjugate in C. In recent unpublished work G.A.Chambers of the University of Wisconsin has proved that, if is a saturated formation, the -normalizers of an A-group are pronormal subgruops; hence, in particular, that two -normalizers contained in an -projector E of an A-group are conjugate in E. In this note we describe an example which shows that in Alperin's theorem the class of nilpotent groups cannot in general be replaced by an arbitary saturated formation without some restriction on the class of soluble groups under consideration. we provePROPOSITION. There exists a saturated formationand a group G which has two-normalizers E1and E2contained in an-projector F of G such that E1and E2are not conjugate in F.

scholarly journals On a relation between injectors and certain complemented chief factors of finite soluble groups

1974 ◽  
Vol 17 (4) ◽  
pp. 385-388
Author(s):  
A. R. Makan
Keyword(s):  
Soluble Group ◽  
Fitting Class ◽  
Chief Factor ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
Chief Factors

The Fitting class of finite soluble π-groups, where π is an arbitrary set of primes, has the property that each complement of an -avoided, complemented chief factor of any finite soluble group G contains an -injector of G. In other words, each -avoided, complemented chief factor of G is -complemented in the sense of Hartley (see [2]).

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