On elements of order 2 in the free centre-by-(nilpotent of class 3)-by-abelian group within the variety of all soluble groups of derived length 3

2021 ◽  
pp. 1-35
Author(s):  
Seyhun Kesim
Keyword(s):  
Abelian Group ◽  
Soluble Groups ◽  
Derived Length

On the derived length of subgroups of infinite index in soluble groups

2003 ◽  
Vol 7 (1) ◽  
Author(s):  
Martyn R. Dixon ◽  
Martin J. Evans ◽  
Howard Smith
Keyword(s):  
Infinite Index ◽  
Soluble Groups ◽  
Derived Length

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 Representations of metabelian groups satisfying the minimal condition for normal subgroups

1976 ◽  
Vol 14 (2) ◽  
pp. 267-278 ◽  
Author(s):  
Howard L. Silcock
Keyword(s):  
Minimal Condition ◽  
Normal Subgroups ◽  
Metabelian Groups ◽  
Soluble Groups ◽  
Derived Length ◽  
Indecomposable Representations

A question of John S. Wilson concerning indecomposable representations of metabelian groups satisfying the minimal condition for normal subgroups is answered negatively, by means of an example. It is shown that such representations need not be irreducible, even when the group being represented is an extension of an elementary abelian p–group by a quasicyclic q–group of the type first described by V.S. Čarin, and the characteristic of the field is a prime distinct from both p and q. This implies that certain techniques used in the study of metabelian groups satisfying the minimal condition for normal subgroups are not available for the corresponding class of soluble groups of derived length 3.

Representations of holomorphs of group extensions with Abelian kernels

1975 ◽  
Vol 78 (3) ◽  
pp. 357-368 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Finite Rank ◽  
Automorphism Groups ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Soluble Groups ◽  
Group Extensions

This paper is devoted to the construction of faithful representations of the automorphism group and the holomorph of an extension of an abelian group by some other group, the representations here being homomorphisms into certain restricted parts of the automorphism groups of smallish abelian groups. We apply these to two very specific cases, namely to finitely generated metabelian groups and to certain soluble groups of finite rank. We describe the applications first.

scholarly journals A note on subnormal defect in finite soluble groups

1989 ◽  
Vol 39 (2) ◽  
pp. 255-258
Author(s):  
R.A. Bryce
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Soluble Groups ◽  
Sylow Subgroups ◽  
Derived Length ◽  
A Finite Group ◽  
Subnormal Subgroups

It is shown that for every positive integer n there exists a finite group of derived length n in which all Sylow subgroups are abeian and in which the defect of subnormal subgroups is at most 3.

scholarly journals Length functions on hypercentral groups

1985 ◽  
Vol 28 (3) ◽  
pp. 303-304
Author(s):  
David L. Wilkens
Keyword(s):  
Abelian Group ◽  
Free Product ◽  
Dihedral Group ◽  
Value Function ◽  
Length Function ◽  
Soluble Groups ◽  
Absolute Value ◽  
Easy Consequence ◽  
Cyclic Groups ◽  
A Chain

In [6] the structure of any real valued length function on an abelian group G is determined. It is shown there, in Theorem 6.1., that such a length function is an extension of a non-Archimedean length function l1 on N by an Archimedean length function l2 on H=G/N. Any non-Archimedean length function is given by a chain of subgroups, as described in [5], and following from results of Nancy Harrison [2], the length l2 is essentially the absolute value function on a subgroup of R. In the situation above if N≠G then N is a subgroup of G whose elements have bounded lengths. In this paper we show that it is an easy consequence of techniques developed in [1] that this result can be extended to hypercentral groups, thus determining the structure of any length function in this case. We point out that the result does not extend to soluble groups. The infinite dihedral group D∞ is soluble. However if D∞ is regarded as a free product of two cyclic groups of order 2 and is given the length function associated with a free product, as described by Lyndon [3], then N is not a subgroup of D∞, and the lengths of its elements are unbounded.

On Soluble Groups of Automorphism of Riemann Surfaces

1991 ◽  
Vol 34 (1) ◽  
pp. 67-73 ◽  
Author(s):  
Grzegorz Gromadzki
Keyword(s):  
Riemann Surface ◽  
Automorphism Group ◽  
Positive Integer ◽  
Riemann Surfaces ◽  
Soluble Group ◽  
Compact Riemann Surface ◽  
Soluble Groups ◽  
Group Of Automorphisms ◽  
Derived Length

AbstractLet G be a soluble group of derived length 3. We show in this paper that if G acts as an automorphism group on a compact Riemann surface of genus g ≠ 3,5,6,10 then it has at most 24(g — 1) elements. Moreover, given a positive integer n we show the existence of a Riemann surface of genus g = n4 + 1 that admits such a group of automorphisms of order 24(g — 1), whilst a surface of specified genus can admit such a group of automorphisms of order 48(g — 1), 40(g — 1), 30(g — 1) and 36(g — 1) respectively.

Soluble groups with few conjugate classes of non-cyclic subgroups

2019 ◽  
Vol 18 (06) ◽  
pp. 1950114 ◽  
Author(s):  
Heng Lv ◽  
Chong Zhao ◽  
Wei Zhou
Keyword(s):  
Nilpotent Group ◽  
Soluble Groups ◽  
Cyclic Subgroups ◽  
Derived Length

For a group [Formula: see text], let [Formula: see text] be the number of conjugate classes of the non-cyclic subgroups. In this paper, we prove that the derived length of the group [Formula: see text] with [Formula: see text] is at most 3, and we also study the non-nilpotent group [Formula: see text] with [Formula: see text].

scholarly journals Constructing uncountably many groups with the same profinite completion

10.53733/89 ◽  
2021 ◽  
Vol 52 ◽  
pp. 765-771
Author(s):  
Nikolay Nikolov ◽  
Dan Segal
Keyword(s):  
Rooted Tree ◽  
The Other ◽  
Profinite Completion ◽  
Soluble Groups ◽  
Derived Length

Two constructions are described: one gives soluble groups of derived length 4, the other uses groups acting on a rooted tree.

A class of deficiency zero soluble groups of derived length 4

1992 ◽  
Vol 121 (1-2) ◽  
pp. 163-168 ◽  
Author(s):  
H. Doostie ◽  
A. R. Jamali
Keyword(s):  
Soluble Groups ◽  
Derived Length

SynopsisIn this paper we study a class of 2-generator 2-relator groups G(m) and show that they are all finite. Moreover, two infinite subclasses are soluble of derived length 4.

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