scholarly journals Bounding the fitting height in terms of the exponent

Author(s):  
Francesco Fumagalli ◽  
Felix Leinen ◽  
Orazio Puglisi
Keyword(s):  
1984 ◽  
Vol 86 (2) ◽  
pp. 555-566 ◽  
Author(s):  
Alexandre Turull
Keyword(s):  

2014 ◽  
Vol 21 (02) ◽  
pp. 355-360
Author(s):  
Xianxiu Zhang ◽  
Guangxiang Zhang

In this article, we prove that a finite solvable group with character degree graph containing at least four vertices has Fitting height at most 4 if each derived subgraph of four vertices has total degree not more than 8. We also prove that if the vertex set ρ(G) of the character degree graph Δ(G) of a solvable group G is a disjoint union ρ(G) = π1 ∪ π2, where |πi| ≥ 2 and pi, qi∈ πi for i = 1,2, and no vertex in π1 is adjacent in Δ(G) to any vertex in π2 except for p1p2 and q1q2, then the Fitting height of G is at most 4.


1990 ◽  
Vol 107 (2) ◽  
pp. 227-238 ◽  
Author(s):  
Alexandre Turull

Let G be a finite solvable group and A a group of automorphisms of G such that (|A|, |G|) = 1. We denote by h(G) the Fitting height of G and by l(A) the length of the longest chain of subgroups of A. Then, under some additional hypotheses, it is known from [5] that h(G) ≤ 2l(A) + h(CG(A)) and from [8] that, when CG(A) = 1, h(G) ≤ l(A), both results being best possible (see [6, 7]). The present paper attempts to explain the difference in the coefficient of l(A) in the two inequalities, from 2 to 1.


2014 ◽  
Vol 17 (5) ◽  
Author(s):  
Carlo Casolo ◽  
Enrico Jabara ◽  
Pablo Spiga

AbstractIn this paper we are concerned with finite soluble groups


1987 ◽  
Vol 102 (3) ◽  
pp. 431-441 ◽  
Author(s):  
Brian Hartley ◽  
Volker Turau

Let G be a finite soluble group with Fitting subgroup F(G). The Fitting series of G is defined, as usual, by F0(G) = 1 and Fi(G)/Fi−1(G) = F(G/Fi−1(G)) for i ≥ 1, and the Fitting height h = h(G) of G is the least integer such that Fn(G) = G. Suppose now that a finite soluble group A acts on G. Let k be the composition length of A, that is, the number of prime divisors (counting multiplicities) of |A|. There is a certain amount of evidence in favour of theCONJECTURE. |G:Fk(G)| is bounded by a number depending only on |A| and |CG(A)|.


2020 ◽  
Vol 30 (05) ◽  
pp. 1073-1080
Author(s):  
Güli̇n Ercan ◽  
İsmai̇l Ş. Güloğlu

Let [Formula: see text] be a finite solvable group and [Formula: see text] be a subgroup of [Formula: see text]. Suppose that there exists an [Formula: see text]-invariant Carter subgroup [Formula: see text] of [Formula: see text] such that the semidirect product [Formula: see text] is a Frobenius group with kernel [Formula: see text] and complement [Formula: see text]. We prove that the terms of the Fitting series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the Fitting series of [Formula: see text], and the Fitting height of [Formula: see text] may exceed the Fitting height of [Formula: see text] by at most one. As a corollary it is shown that for any set of primes [Formula: see text], the terms of the [Formula: see text]-series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the [Formula: see text]-series of [Formula: see text], and the [Formula: see text]-length of [Formula: see text] may exceed the [Formula: see text]-length of [Formula: see text] by at most one. These theorems generalize the main results in [E. I. Khukhro, Fitting height of a finite group with a Frobenius group of automorphisms, J. Algebra 366 (2012) 1–11] obtained by Khukhro.


1987 ◽  
Vol 30 (1) ◽  
pp. 51-56 ◽  
Author(s):  
Cheng Kai-Nah

By the results of Rickman [7] and Ralston [6], a finite group G admitting a fixed point free automorphism α of order pq, where p and q are primes, is soluble. If p = q, then |G| is necessarily coprime to |α|, and it follows from Berger [1] that G has Fitting height at most 2, the composition length of <α>. The purpose of this paper is to prove a corresponding result in the case when p≠q.


2013 ◽  
Vol 41 (8) ◽  
pp. 2869-2878 ◽  
Author(s):  
Claudio Paolo Morresi Zuccari

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