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.

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 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 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 The subnormal structure of some classes of coluble groups

1972 ◽  
Vol 13 (3) ◽  
pp. 365-377 ◽  
Author(s):  
D. McDougall
Keyword(s):  
Finite Groups ◽  
Soluble Group ◽  
Subnormal Subgroup ◽  
Negative Integer ◽  
Soluble Groups ◽  
Derived Length ◽  
Subnormal Structure ◽  
A Chain

Finite groups in which normality is transitive have been studied by Best and Taussky, [1], Gaschütz, [3], and Zacher [16]. Infinite soluble groups in which normality is transitive have been studied by Robinson in [9]. A subgroup H of a group G is subnormal in G if H can be connected to G by a chain of r subgroups, in which each is normal in its successor, where r is a non-negative integer. The least such r is called the subnormal index of H in G (or the defect of H in G). Then groups in which normality is transitive are precisely those in which every subnormal subgroup has subnormal index at most one. Thus the structure of soluble groups in which every subnormal subgroup has subnormal index at most n (such a group is said to have bounded subnormal indices) has been dealt with by Robinson in [9] for the case where n is one. However Theorem D of [12] states that a soluble group of derived length n can be embedded in a soluble group in which the subnormal indices are at most n. Therefore we must impose further conditions on the groups if we hope to obtain any worthwhile results for the above problem with n greater than one.

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.

ON A PERMUTABILITY PROPERTY OF SUBGROUPS OF FINITE SOLUBLE GROUPS

2010 ◽  
Vol 12 (02) ◽  
pp. 207-221 ◽  
Author(s):  
A. BALLESTER-BOLINCHES ◽  
JOHN COSSEY ◽  
X. SOLER-ESCRIVÀ
Keyword(s):  
Soluble Group ◽  
Subnormal Subgroup ◽  
Finite Soluble Group ◽  
Supersoluble Groups ◽  
Soluble Groups ◽  
Sylow Subgroups ◽  
Subnormal Subgroups

The structure and embedding of subgroups permuting with the system normalizers of a finite soluble group are studied in the paper. It is also proved that the class of all finite soluble groups in which every subnormal subgroup permutes with the Sylow subgroups is properly contained in the class of all soluble groups whose subnormal subgroups permute with the system normalizers while this latter is properly contained in the class of all supersoluble groups.

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].

scholarly journals Bounds on the fitting length of finite soluble groups with supersoluble Sylow normalisers

1991 ◽  
Vol 44 (1) ◽  
pp. 19-31 ◽  
Author(s):  
R.A. Bryce ◽  
V. Fedri ◽  
L. Serena
Keyword(s):  
Soluble Group ◽  
Chief Factor ◽  
Structural Constraints ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
Fitting Length

We prove that, in a finite soluble group, all of whose Sylow normalisers are super-soluble, the Fitting length is at most 2m + 2, where pm is the highest power of the smallest prime p dividing |G/Gs| here Gs is the supersoluble residual of G. The bound 2m + 2 is best possible. However under certain structural constraints on G/GS, typical of the small examples one makes by way of experimentation, the bound is sharply reduced. More precisely let p be the smallest, and r the largest, prime dividing the order of a group G in the class under consideration. If a Sylow p–subgroup of G/GS acts faithfully on every r-chief factor of G/GS, then G has Fitting length at most 3.

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.

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