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].

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 On varieties of soluble groups II

1972 ◽  
Vol 7 (3) ◽  
pp. 437-441 ◽  
Author(s):  
J.R.J. Groves
Keyword(s):  
Nilpotent Group ◽  
Metabelian Groups ◽  
Locally Finite ◽  
Soluble Groups ◽  
Finite Exponent

It is shown that, in a variety which does not contain all metabelian groups and is contained in a product of (finitely many) varieties each of which is soluble or locally finite, every group is an extension of a group of finite exponent by a nilpotent group by a group of finite exponent.

The derived length of the unit group of a group algebra — The case G′ = Sylp(G)

2016 ◽  
Vol 16 (08) ◽  
pp. 1750142 ◽  
Author(s):  
Tibor Juhász
Keyword(s):  
Nilpotent Group ◽  
Group Algebra ◽  
Upper Bound ◽  
Commutator Subgroup ◽  
Unit Group ◽  
Derived Length ◽  
Group Of Units

Let [Formula: see text] be an odd prime, and let [Formula: see text] be a nilpotent group, whose commutator subgroup is finite abelian satisfying [Formula: see text] and [Formula: see text]. In this contribution, an upper bound is given on the derived length of the group of units of the group algebra of [Formula: see text] over a field of characteristic [Formula: see text]. Furthermore, we show that this bound is achieved, whenever [Formula: see text] is cyclic.

scholarly journals Groups with many permutable subgroups

1990 ◽  
Vol 48 (3) ◽  
pp. 397-401 ◽  
Author(s):  
Mario Curzio ◽  
John Lennox ◽  
Akbar Rhemtulla ◽  
James Wiegold
Keyword(s):  
Free Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Cyclic Subgroups ◽  
Prove Theorem ◽  
Permutable Subgroups ◽  
Infinite Set ◽  
Torsion Free

AbstractWe consider the influence on a group G of the condition that every infinite set of cyclic subgroups of G contains a pair that permute and prove (Theorem 1) that finitely generated soluble groups with this condition are centre-by-finite, and (Theorem 2) that torsion free groups satisfying the condition are abelian.

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.

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.

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.

Wreath products and p–groups

1959 ◽  
Vol 55 (3) ◽  
pp. 224-231 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Nilpotent Group ◽  
Wreath Product ◽  
Soluble Group ◽  
Nilpotent Groups ◽  
Wreath Products ◽  
Soluble Groups ◽  
Finite Exponent

The wreath product is a useful method for constructing new soluble groups from given ones (cf. P. Hall (3)). Now although the wreath product of one soluble group by another is (obviously) always soluble, the corresponding result is no longer true for nilpotent groups. It is the object of § 3 of this note to determine precisely when the wreath product W of a non-trivial nilpotent group A by a non-trivial nilpotent group B is nilpotent; in fact I prove that W is nilpotent if and only if both A and B are (nilpotent) p–groups with A of finite exponent and B finite.

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.

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