SOME CRITERIA FOR EXISTENCE OF SUPPLEMENTS TO NORMAL SUBGROUPS AND THEIR APPLICATIONS

2010 ◽  
Vol 20 (05) ◽  
pp. 689-719 ◽  
Author(s):  
LEONID A. KURDACHENKO ◽  
JAVIER OTAL ◽  
IGOR YA. SUBBOTIN
Keyword(s):  
Normal Subgroups ◽  
Finitely Generated ◽  
Original Proof ◽  
Soluble Groups ◽  
Finite Section ◽  
Abelian Subgroups ◽  
Section Rank ◽  
The Many ◽  
New Criteria

We established several new criteria for existence of complements and supplements to some normal abelian subgroups in groups. In passing, as one of the many useful applications and corollaries of these results, we obtained a description of some finitely generated soluble groups of finite Hirsch–Zaitsev rank. As another application of our results, we obtained a D.J.S. Robinson's theorem on structure of finitely generated soluble groups of finite section rank. The original proof of this theorem was homological, but all proofs in this paper, including this one, are purely group-theoretical.

On certain soluble groups

1969 ◽  
Vol 66 (1) ◽  
pp. 1-4 ◽  
Author(s):  
C. K. Gupta
Keyword(s):  
Normal Subgroups ◽  
Finitely Generated ◽  
Maximal Condition ◽  
Metabelian Groups ◽  
Soluble Groups ◽  
Finitely Generated Groups ◽  
The Law

In (2), Hall considered the question: for what varieties of soluble groups do all finitely generated groups satisfy max-n (the maximal condition for normal subgroups)? He has shown that the variety M of metabelian groups and more generally the variety of Abelian-by-nilpotent-of-class-c (c ≥ 1) groups has this property; whereas on the contrary, there are finitely generated groups in the variety V of centre-by-metabelian groups (i.e. defined by the law [x, y; u, v; z]) which do not satisfy max-n. One naturally raises the question: for what subvarieties of V do all finitely generated groups satisfy max-n?

scholarly journals Soluble groups in which every finitely generated subgroup is finitely presented

1978 ◽  
Vol 26 (1) ◽  
pp. 115-125 ◽  
Author(s):  
J. R. J. Groves
Keyword(s):  
Subject Classification ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Maximal Condition ◽  
Soluble Groups ◽  
Image Position ◽  
Finitely Presented

AbstractThe class of finitely generated soluble coherent groups is considered. It is shown that these groups have the maximal condition on normal subgroups and can be characterized in a number of ways. In particular, they are precisely the class of finitely generated soluble groups G with the property:Subject classification (Amer. Math. Soc. (MOS) 1970): primary 20 E 15; secondary 20 F 05.

On R0-Closed Classes, and Finitely Generated Groups

1970 ◽  
Vol 22 (1) ◽  
pp. 176-184 ◽  
Author(s):  
Rex Dark ◽  
Akbar H. Rhemtulla
Keyword(s):  
Normal Subgroups ◽  
Finitely Generated ◽  
Maximal Condition ◽  
Soluble Groups ◽  
Finitely Generated Group ◽  
Finitely Generated Groups ◽  
Central Factors ◽  
Subnormal Subgroups

1.1. If a group satisfies the maximal condition for normal subgroups, then all its central factors are necessarily finitely generated. In [2], Hall asked whether there exist finitely generated soluble groups which do not satisfy the maximal condition for normal subgroups but all of whose central factors are finitely generated. We shall answer this question in the affirmative. We shall also construct a finitely generated group all of whose subnormal subgroups are perfect (and which therefore has no non-trivial central factors), but which does not satisfy the maximal condition for normal subgroups. Related to these examples is the question of which classes of finitely generated groups satisfy the maximal condition for normal subgroups. A characterization of such classes has been obtained by Hall, and we shall include his result as our first theorem.

scholarly journals On F-Normal Subgroups of Finite Soluble Groups

1995 ◽  
Vol 171 (1) ◽  
pp. 189-203 ◽  
Author(s):  
A. Ballesterbolinches ◽  
K. Doerk ◽  
M.D. Perezramos
Keyword(s):  
Normal Subgroups ◽  
Soluble Groups

scholarly journals Locally soluble groups with min-n.

1974 ◽  
Vol 17 (3) ◽  
pp. 305-318 ◽  
Author(s):  
H. Heineken ◽  
J. S. Wilson
Keyword(s):  
Soluble Group ◽  
Free Groups ◽  
Torsion Group ◽  
Minimal Condition ◽  
Normal Subgroups ◽  
Soluble Groups ◽  
Isomorphism Classes ◽  
Torsion Groups ◽  
Torsion Free ◽  
General Method

It was shown by Baer in [1] that every soluble group satisfying Min-n, the minimal condition for normal subgroups, is a torsion group. Examples of non-soluble locally soluble groups satisfying Min-n have been known for some time (see McLain [2]), and these examples too are periodic. This raises the question whether all locally soluble groups with Min-n are torsion groups. We prove here that this is not the case, by establishing the existence of non-trivial locally soluble torsion-free groups satisfying Min-n. Rather than exhibiting one such group G, we give a general method for constructing examples; the reader will then be able to see that a variety of additional conditions may be imposed on G. It will follow, for instance, that G may be a Hopf group whose normal subgroups are linearly ordered by inclusion and are all complemented in G; further, that the countable groups G with these properties fall into exactly isomorphism classes. Again, there are exactly isomorphism classes of countable groups G which have hypercentral nonnilpotent Hirsch-Plotkin radical, and which at the same time are isomorphic to all their non-trivial homomorphic images.

scholarly journals Generating infinite monoids of cellular automata

2021 ◽  
pp. 2250215
Author(s):  
Alonso Castillo-Ramirez
Keyword(s):  
Cellular Automata ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Group Of Units ◽  
Finite Dimensional ◽  
Index Condition ◽  
Indicable Group

For a group [Formula: see text] and a set [Formula: see text], let [Formula: see text] be the monoid of all cellular automata over [Formula: see text], and let [Formula: see text] be its group of units. By establishing a characterization of surjunctive groups in terms of the monoid [Formula: see text], we prove that the rank of [Formula: see text] (i.e. the smallest cardinality of a generating set) is equal to the rank of [Formula: see text] plus the relative rank of [Formula: see text] in [Formula: see text], and that the latter is infinite when [Formula: see text] has an infinite decreasing chain of normal subgroups of finite index, condition which is satisfied, for example, for any infinite residually finite group. Moreover, when [Formula: see text] is a vector space over a field [Formula: see text], we study the monoid [Formula: see text] of all linear cellular automata over [Formula: see text] and its group of units [Formula: see text]. We show that if [Formula: see text] is an indicable group and [Formula: see text] is finite-dimensional, then [Formula: see text] is not finitely generated; however, for any finitely generated indicable group [Formula: see text], the group [Formula: see text] is finitely generated if and only if [Formula: see text] is finite.

scholarly journals Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sam Shepherd ◽  
Daniel J. Woodhouse
Keyword(s):  
Free Group ◽  
Large Family ◽  
Finitely Generated ◽  
Free Vertex ◽  
Cyclic Subgroups ◽  
Graphs Of Groups ◽  
Jsj Decomposition ◽  
Finitely Generated Groups ◽  
Hnn Extensions ◽  
Abelian Subgroups

Abstract We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let G be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that are not quadratically hanging. Our main result is that any group quasi-isometric to G is abstractly commensurable to G. In particular, our result applies to certain “generic” HNN extensions of a free group over cyclic subgroups.

scholarly journals Finitely Generated Soluble Groups and Their Subgroups

2013 ◽  
Vol 41 (5) ◽  
pp. 1790-1799
Author(s):  
Tara Brough ◽  
Derek F. Holt
Keyword(s):  
Finitely Generated ◽  
Soluble Groups

scholarly journals Groups whose irreducible representations have finite degree II

1982 ◽  
Vol 25 (3) ◽  
pp. 237-243 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Characteristic Zero ◽  
Finite Dimension ◽  
Irreducible Representations ◽  
Finitely Generated ◽  
Finite Degree ◽  
Commutative Field ◽  
Soluble Groups

If F is a (commutative) field let denote the class of all groups G such that every irreducible FG-module has finite dimension over F. The introduction to [7] contains motivation for considering these classes and surveys some of the results to date concerning them. In [7] for every field F we determined the finitely generated soluble groups in . Here, for fields F of characteristic zero, we determine, at least in principle, the soluble groups in . Our main result is the following.

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