On the number of conjugacy classes of maximal subgroups in a finite soluble group

1999 ◽  
Vol 72 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Burkhard H�fling
Keyword(s):  
Soluble Group ◽  
Conjugacy Classes ◽  
Maximal Subgroups ◽  
Finite Soluble Group

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.

scholarly journals Enumerating subgroups

1987 ◽  
Vol 43 (2) ◽  
pp. 220-223 ◽  
Author(s):  
R. J. Cook ◽  
James Wiecold ◽  
A. G. Wellamson
Keyword(s):  
Prime Divisor ◽  
Soluble Group ◽  
Maximal Subgroups ◽  
Finite Soluble Group

AbstractIt is proved that a finite soluble group of order n has at most (n − 1)/(q − 1) maximal subgroups, where q is the smallest prime divisor of n.

scholarly journals The Projective Character Tables of a Solvable Group 26:6×2

2019 ◽  
Vol 2019 ◽  
pp. 1-15
Author(s):  
Abraham Love Prins
Keyword(s):  
Simple Group ◽  
Soluble Group ◽  
Galois Field ◽  
Maximal Subgroups ◽  
Finite Soluble Group ◽  
Character Tables ◽  
Extension Group ◽  
Projective Character ◽  
Ordinary Character ◽  
Clifford Theory

The Chevalley–Dickson simple group G24 of Lie type G2 over the Galois field GF4 and of order 251596800=212.33.52.7.13 has a class of maximal subgroups of the form 24+6:A5×3, where 24+6 is a special 2-group with center Z24+6=24. Since 24 is normal in 24+6:A5×3, the group 24+6:A5×3 can be constructed as a nonsplit extension group of the form G¯=24·26:A5×3. Two inertia factor groups, H1=26:A5×3 and H2=26:6×2, are obtained if G¯ acts on 24. In this paper, the author presents a method to compute all projective character tables of H2. These tables become very useful if one wants to construct the ordinary character table of G¯ by means of Fischer–Clifford theory. The method presented here is very effective to compute the irreducible projective character tables of a finite soluble group of manageable size.

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

2006 ◽  
Vol 13 (01) ◽  
pp. 1-8
Author(s):  
Alireza Jamali ◽  
Hamid Mousavi
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Soluble Group ◽  
Maximal Subgroups ◽  
Finite Soluble Group ◽  
A Finite Group

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].

scholarly journals Hypercentral injectors in infinite soluble groups

1979 ◽  
Vol 22 (3) ◽  
pp. 191-194 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Finite Index ◽  
Soluble Group ◽  
Conjugacy Classes ◽  
Finite Soluble Group ◽  
Covering Property ◽  
Soluble Groups ◽  
Finite Case

The Carter subgroups of a finite soluble group may be characterised either as theself-normalising nilpotent subgroups or as the nilpotent projectors. Subgroups with properties analogous to both of these have been considered by Newell (2, 3) in the class of -groups. The results obtained are necessarily less satisfactory than in the finite case, the subgroups either being almost self-normalising (i.e. having finite index in their normaliser) or having an almost-covering property. Also the subgroups are not necessarily conjugate but lie in finitely many conjugacy classes.

scholarly journals On a relation between the Fitting length of a soluble group and the number of conjugacy classes of its maximal nilpotent subgroups

1969 ◽  
Vol 1 (1) ◽  
pp. 3-10 ◽  
Author(s):  
H. Lausch ◽  
A. Makan
Keyword(s):  
Soluble Group ◽  
Conjugacy Classes ◽  
Finite Soluble Group ◽  
Prime Divisors ◽  
Soluble Groups ◽  
Odd Order ◽  
Fitting Length ◽  
Qualitative Nature ◽  
Positive Integers

In a finite soluble group G, the Fitting (or nilpotency) length h(G) can be considered as a measure for how strongly G deviates from being nilpotent. As another measure for this, the number v(G) of conjugacy classes of the maximal nilpotent subgroups of G may be taken. It is shown that there exists an integer-valued function f on the set of positive integers such that h(G) ≦ f(v(G)) for all finite (soluble) groups of odd order. Moreover, if all prime divisors of the order of G are greater than v(G)(v(G) - l)/2, then h(G) ≦3. The bound f(v(G)) is just of qualitative nature and by far not best possible. For v(G) = 2, h(G) = 3, some statements are made about the structure 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.

scholarly journals The Fitting length of a finite soluble group and the number of conjugacy classes of its maximal nilpotent subgroups

1972 ◽  
Vol 6 (2) ◽  
pp. 213-226 ◽  
Author(s):  
A.R. Makan
Keyword(s):  
Upper Bound ◽  
Soluble Group ◽  
Conjugacy Classes ◽  
Finite Soluble Group ◽  
Fitting Length

It is shown that there exists a logarithmic upper bound on the Fitting length h(G) of a finite soluble group G in terms of the number ν(G) of the conjugacy classes of its maximal nilpotent subgroups. For ν(G) = 3, the best possible bound on h(G) is shown to be 4.

The Fitting Length of a Finite Soluble Group and the Number of Conjugacy Classes of its Maximal Metanilpotent Subgroups

1973 ◽  
Vol 16 (2) ◽  
pp. 233-237
Author(s):  
A. R. Makan
Keyword(s):  
Soluble Group ◽  
Conjugacy Classes ◽  
Finite Soluble Group ◽  
Fitting Length ◽  
Prime 2

It is known that the Fitting length h(G) of a finite soluble group G is bounded in terms of the number v(G) of the conjugacy classes of its maximal nilpotent subgroups. For |G| odd, a bound on h(G) in terms of v(G) was discussed in Lausch and Makan [6]. In the case when the prime 2 divides |G|, a logarithmic bound on h(G) in terms of v(G) is obtained in [7]. The main purpose of this paper is to show that the Fitting length of a finite soluble group is also bounded in terms of the number of conjugacy classes of its maximal metanilpotent subgroups. In fact, our result is rather more general.

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