scholarly journals On the theory of soluble factorizable groups

1976 ◽  
Vol 15 (1) ◽  
pp. 97-110 ◽  
Author(s):  
Otto-Uwe Kramer
Keyword(s):  
Soluble Group ◽  
Finite Soluble Group ◽  
General Estimate ◽  
Nilpotent Length ◽  
Derived Length ◽  
Odd Order

Suppose that a finite soluble group G is the product AB of subgroups A and B. Our question is the following: what conclusions can be made about G if A and B are suitably restricted? First we shall prove that the p–length of G is restricted by the derived lengths of the Sylow p–subgroups of A and B, if A and B are p–closed and p′-closed. Moreover, if in such a group the Sylow p–subgroups of A and B are modular, the p–length of G is at most 1. Next we obtain a general estimate for the derived length of the group G = AB of odd order in terms of the derived lengths of A and B. Furthermore it will be possible to bound the nilpotent length of G and also the p–length of G in terms of other invariants of special subgroups of G.

scholarly journals Groups of odd order in which every subnormal subgroup has defect at most two

1991 ◽  
Vol 51 (2) ◽  
pp. 331-342
Author(s):  
John Cossey
Keyword(s):  
Soluble Group ◽  
Subnormal Subgroup ◽  
Finite Soluble Group ◽  
Derived Length ◽  
Odd Order ◽  
Fitting Length

AbstractIn 1980, McCaughan and Stonehewer showed that a finite soluble group in which every subnormal subgroup has defect at most two has derived length at most nine and Fitting length at most five, and gave an example of derived length five and Fitting length four. In 1984 Casolo showed that derived length five and Fitting length four are best possible bounds.In this paper we show that for groups of odd order the bounds can be improved. A group of odd order with every subnormal subgroup of defect at most two has derived and Fitting length at most three, and these bounds are best possible.

On cofactors of subnormal subgroups

2016 ◽  
Vol 15 (09) ◽  
pp. 1650169
Author(s):  
Victor Monakhov ◽  
Irina Sokhor
Keyword(s):  
Finite Group ◽  
Soluble Group ◽  
Upper Bounds ◽  
Finite Soluble Group ◽  
Nilpotent Length ◽  
Derived Length ◽  
Text Length ◽  
Subnormal Subgroups

For a soluble finite group [Formula: see text] and a prime [Formula: see text] we let [Formula: see text], [Formula: see text]. We obtain upper bounds for the rank, the nilpotent length, the derived length, and the [Formula: see text]-length of a finite soluble group [Formula: see text] in terms of [Formula: see text] and [Formula: see text].

Solvable groups with restrictions on Sylow subgroups of the Fitting subgroup

2016 ◽  
Vol 09 (02) ◽  
pp. 1650037
Author(s):  
Alexander Trofimuk
Keyword(s):  
Free Groups ◽  
Solvable Groups ◽  
Fitting Subgroup ◽  
Sylow Subgroups ◽  
Nilpotent Length ◽  
Derived Length ◽  
Odd Order

In this paper, we study solvable groups in which [Formula: see text] is at most 2. In particular, we investigated groups of odd order and [Formula: see text]-free groups with this property. Exact estimations of the derived length and nilpotent length of such groups are obtained.

scholarly journals Finite groups of deficiency zero involving the Lucas numbers

1990 ◽  
Vol 33 (1) ◽  
pp. 1-10 ◽  
Author(s):  
C. M. Campbell ◽  
E. F. Robertson ◽  
R. M. Thomas
Keyword(s):  
Finite Groups ◽  
Soluble Group ◽  
Fibonacci Number ◽  
Single Parameter ◽  
The Other ◽  
Finite Soluble Group ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Derived Length ◽  
New Class

In this paper, we investigate a class of 2-generator 2-relator groups G(n) related to the Fibonacci groups F(2,n), each of the groups in this new class also being defined by a single parameter n, though here n can take negative, as well as positive, values. If n is odd, we show that G(n) is a finite soluble group of derived length 2 (if n is coprime to 3) or 3 (otherwise), and order |2n(n + 2)gnf(n, 3)|, where fn is the Fibonacci number defined by f0=0,f1=1,fn+2=fn+fn+1 and gn is the Lucas number defined by g0 = 2, g1 = 1, gn+2 = gn + gn+1 for n≧0. On the other hand, if n is even then, with three exceptions, namely the cases n = 2,4 or –4, G(n) is infinite; the groups G(2), G(4) and G(–4) have orders 16, 240 and 80 respectively.

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.

scholarly journals The nilpotent length of finite soluble groups

1973 ◽  
Vol 16 (3) ◽  
pp. 357-362 ◽  
Author(s):  
M. Frick
Keyword(s):  
Soluble Group ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
Nilpotent Length

Throughout this paper a “group” will mean a “finite soluble group”.

scholarly journals On soluble groups which admit the dihedral group of order eight fixed-point-freely

1973 ◽  
Vol 8 (2) ◽  
pp. 305-312 ◽  
Author(s):  
Alan R. Camina ◽  
F. Peter Lockett
Keyword(s):  
Fixed Point ◽  
Free Group ◽  
Soluble Group ◽  
Dihedral Group ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
Group Of Automorphisms ◽  
Nilpotent Length

If the finite soluble group G admits the dihedral group of order eight as a fixed-point-free group of automorphisms then the nilpotent length of G is at most three.

scholarly journals ON SOLUBILITY OF GROUPS WITH BOUNDED CENTRALIZER CHAINS

2009 ◽  
Vol 51 (1) ◽  
pp. 49-54 ◽  
Author(s):  
E. I. KHUKHRO
Keyword(s):  
Soluble Group ◽  
Maximum Length ◽  
Finite Soluble Group ◽  
Derived Length ◽  
A Chain ◽  
Locally Soluble Group

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.

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.

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