scholarly journals On nilpotency of higher commutator subgroups of a finite soluble group

2020 ◽  
Author(s):  
Josean da Silva Alves ◽  
Pavel Shumyatsky
Keyword(s):  
Soluble Group ◽  
Finite Soluble Group

A Class of Three-Generator, Three-Relation, Finite Groups

1970 ◽  
Vol 22 (1) ◽  
pp. 36-40 ◽  
Author(s):  
J. W. Wamsley
Keyword(s):  
Nilpotent Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Finite Soluble Group ◽  
Finite Nilpotent Group ◽  
Image Position

Mennicke (2) has given a class of three-generator, three-relation finite groups. In this paper we present a further class of three-generator, threerelation groups which we show are finite.The groups presented are defined as:with α|γ| ≠ 1, β|γ| ≠ 1, γ ≠ 0.We prove the following result.THEOREM 1. Each of the groups presented is a finite soluble group.We state the following theorem proved by Macdonald (1).THEOREM 2. G1(α, β, 1) is a finite nilpotent group.1. In this section we make some elementary remarks.

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 Wielandt Subgroup of a Finite Soluble Group

1989 ◽  
Vol s2-40 (2) ◽  
pp. 244-256 ◽  
Author(s):  
R. A. Bryce ◽  
John Cossey
Keyword(s):  
Soluble Group ◽  
Finite Soluble Group

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 Subgroups like Wielandt's in soluble groups

2000 ◽  
Vol 42 (1) ◽  
pp. 67-74 ◽  
Author(s):  
Clara Franchi
Keyword(s):  
Soluble Group ◽  
Subject Classification ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
Subnormal Subgroups

For each m≥1, u_{m}(G) is defined to be the intersection of the normalizers of all the subnormal subgroups of defect at most m in G. An ascending chain of subgroups u_{m,i}(G) is defined by setting u_{m,i}(G)/u_{m,i−1}(G)=u_{m}(G/u_{m,i−1}(G)). If u_{m,n}(G)=G, for some integer n, the m-Wielandt length of G is the minimal of such n.In [3], Bryce examined the structure of a finite soluble group with given m-Wielandt length, in terms of its polynilpotent structure. In this paper we extend his results to infinite soluble groups.1991 Mathematics Subject Classification. 20E15, 20F22.

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.

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