On soluble groups which admit the dihedral group of order eight fixed-point-freely (original) (raw)
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The nilpotent length of finite soluble groups
Journal of the Australian Mathematical Society, 1973
References Chapter 1. Preliininaries 1.1. General results. The Frattini subgroup of a group G, denoted by <I'(G), is the intersection of all the maximal subgroups of G. An element ^G is said to be omissible in G if, whenever <g,X> = G for some subset X of G, then G =<x). We now state som.e well-known results concerning the Frattini subgroup, v;hich will be used frequently in the sequel. The proofs of these results can all be found in Section 3.3 of [7]. Lemma 1.1.1 (a) An element x of G lies in ^ (.G) if and only if x is omissible in G. (b) If N G then G has a proper subgroup H such that NH = G if and only if N ^ 9(G). (c) The Frattini subgroup of a group is nilpotent. (d) If G has a normal subgroup N ^ (G) such that G/N is nilpotent, then G is itself nilpotent.
Bulletin of the Australian Mathematical Society, 1969
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 the p-length of some finite p-soluble groups
Israel Journal of Mathematics, 2014
The main aim of this paper is to give structural information of a finite group of minimal order belonging to a subgroup-closed class of finite groups and whose p-length is greater than 1, p a prime number. Alternative proofs and improvements of recent results about the influence of minimal p-subgroups on the p-nilpotence and p-length of a finite group arise as consequences of our study.
On the influence of fixed point free nilpotent automorphism groups
Monatshefte für Mathematik, 2016
A finite group F H is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that [F, h] = F for all nonidentity elements h ∈ H. Let F H be a Frobenius-like group with complement H of prime order such that C F (H) is of prime order. Suppose that F H acts on a finite group G by automorphisms where (|G|, |H |) = 1 in such a way that C G (F) = 1. In the present paper we prove that the Fitting series of C G (H) coincides with the intersections of C G (H) with the Fitting series of G, and the nilpotent length of G exceeds the nilpotent length of C G (H) by at most one. As a corollary, we also prove that for any set of primes π , the upper π-series of C G (H) coincides with the intersections of C G (H) with the upper π-series of G, and the π-length of G exceeds the π-length of C G (H) by at most one. Keywords Frobenius-like group • Fixed points • Nilpotent length • π-length Mathematics Subject Classification 20D10 • 20D15 • 20D45 Communicated by J. S. Wilson.
Bulletin of the Australian Mathematical Society, 1972
The least upper bound for the nilpotent lengths of soluble linear groups of degree n is calculated. For each n it is k + 2r where r(n) = [log_(2«-l)A] and [x] is the integral part of
The composition and derived lengths of a soluble group
Journal of Algebra, 1989
Given a finite soluble group with derived length d and composition length n, the present paper investigates upper bounds for d in terms of n. An elementary argument is used to show that d< r2n/31, where [2n/31 denotes the least integer greater than or equal to 2n/3. The sharper bound d<r(n+ 3)/2-3/(n+2)1 is obtained by using properties of soluble subgroups of two-dimensional general linear groups. Finally, arguments like those used by Hall and Higman are used in conjunction with bounds for the derived length of soluble linear groups to show that d < f(n) < 3 log,n + 9.
Nilpotency and solubility of groups relative to an automorphism
In this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. With this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism α are defined. N (G) and S(G) are the set of all nilpotency classes and the set of all solvability classes for the group G with respect to different automorphisms of the group, respectively. If G is nilpotent or solvable with respect to the all its automorphisms, then is referred as it absolute nilpotent or solvable group. Subsequently, N (G) and S(G) are obtained for certain groups. This work is a study of the nilpotency and solvability of the group G from the point of view of the automorphism which the nilpotent and solvable groups have been divided to smaller classes of the nilpotency and the solvability with respect to its automorphisms.
A note on p-nilpotence and solvability of finite groups
Journal of Algebra, 2009
In this note, we first give some examples to show that some hypotheses of some well-known results for a group G to be pnilpotent, solvable and supersolvable are essential and cannot be removed. Second, we give some generalizations of two theorems in [A. Ballester-Bolinches, X. Guo, Some results on p-nilpotence and solubility of finite groups, J. Algebra 228 (2000) 491-496].