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.

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.

scholarly journals Groups with a quotient that contains the original group as a direct factor

1992 ◽  
Vol 45 (3) ◽  
pp. 513-520 ◽  
Author(s):  
Ron Hirshon ◽  
David Meier
Keyword(s):  
Free Group ◽  
Normal Subgroups ◽  
Direct Factor ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Finitely Generated Groups ◽  
Original Group ◽  
Hopfian Group ◽  
Finitely Presented

We prove that given a finitely generated group G with a homomorphism of G onto G × H, H non-trivial, or a finitely generated group G with a homomorphism of G onto G × G, we can always find normal subgroups N ≠ G such that G/N ≅ G/N × H or G/N ≅ G/N × G/N respectively. We also show that given a finitely presented non-Hopfian group U and a homomorphism φ of U onto U, which is not an isomorphism, we can always find a finitely presented group H ⊇ U and a finitely generated free group F such that φ induces a homomorphism of U * F onto (U * F) × H. Together with the results above this allows the construction of many examples of finitely generated groups G with G ≅ G × H where H is finitely presented. A finitely presented group G with a homomorphism of G onto G × G was first constructed by Baumslag and Miller. We use a slight generalisation of their method to obtain more examples of such groups.

scholarly journals The subnormal coalescence of some classes of groups of finite rank

1973 ◽  
Vol 16 (3) ◽  
pp. 324-327 ◽  
Author(s):  
Mark Drukker ◽  
Derek J. S. Robinson ◽  
Ian Stewart
Keyword(s):  
Finite Groups ◽  
Finite Rank ◽  
Nilpotent Groups ◽  
Subnormal Subgroup ◽  
Minimal Condition ◽  
Finitely Generated ◽  
Maximal Condition ◽  
Finitely Generated Groups ◽  
Subnormal Subgroups

A class of groups forms a (subnormal) coalition class, or is (subnormally) coalescent, if wheneverHandKare subnormal -subgroups of a groupGthen their join <H, K> is also a subnormal -subgroup ofG. Among the known coalition classes are those of finite groups and polycylic groups (Wielandt [15]); groups with maximal condition for subgroups (Baer [1]); finitely generated nilpotent groups (Baer [2]); groups with maximal or minimal condition on subnormal subgroups (Robinson [8], Roseblade [11, 12]); minimax groups (Roseblade, unpublished); and any subjunctive class of finitely generated groups (Roseblade and Stonehewer [13]).

On characteristically simple groups

1976 ◽  
Vol 80 (1) ◽  
pp. 19-35 ◽  
Author(s):  
J. S. Wilson
Keyword(s):  
Simple Groups ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Maximal Condition ◽  
Finitely Generated Groups

1·1. A group is called characteristically simple if it has no proper non-trivial subgroups which are left invariant by all of its automorphisms. One familiar class of characteristically simple groups consists of all direct powers of simple groups: this contains all finite characteristically simple groups, and, more generally, all characteristically simple groups having minimal normal subgroups. However not all characteristically simple groups lie in this class because, for instance, additive groups of fields are characteristically simple. Our object here is to construct finitely generated groups, and also groups satisfying the maximal condition for normal subgroups, which are characteristically simple but which are not direct powers of simple groups.

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.

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.

A Conjecture of Bachmuth and Mochizuki on Automorphisms of Soluble Groups

1976 ◽  
Vol 28 (6) ◽  
pp. 1302-1310 ◽  
Author(s):  
Brian Hartley
Keyword(s):  
Finite Index ◽  
Soluble Group ◽  
Abelian Groups ◽  
Free Subgroup ◽  
Linear Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Group Of Automorphisms ◽  
Finitely Generated Group ◽  
Soluble Subgroup

In [1], Bachmuth and Mochizuki conjecture, by analogy with a celebrated result of Tits on linear groups [8], that a finitely generated group of automorphisms of a finitely generated soluble group either contains a soluble subgroup of finite index (which may of course be taken to be normal) or contains a non-abelian free subgroup. They point out that their conjecture holds for nilpotent-by-abelian groups and in some other cases.

Context-sensitive languages and G-automata

2017 ◽  
Vol 27 (02) ◽  
pp. 237-249
Author(s):  
Rachel Bishop-Ross ◽  
Jon M. Corson ◽  
James Lance Ross
Keyword(s):  
Word Problem ◽  
Finitely Generated ◽  
Closure Properties ◽  
Finitely Generated Group ◽  
Context Sensitive ◽  
The Family

For a given finitely generated group [Formula: see text], the type of languages that are accepted by [Formula: see text]-automata is determined by the word problem of [Formula: see text] for most of the classical types of languages. We observe that the only exceptions are the families of context-sensitive and recursive languages. Thus, in general, to ensure that the language accepted by a [Formula: see text]-automaton is in the same classical family of languages as the word problem of [Formula: see text], some restriction must be imposed on the [Formula: see text]-automaton. We show that restricting to [Formula: see text]-automata without [Formula: see text]-transitions is sufficient for this purpose. We then define the pullback of two [Formula: see text]-automata and use this construction to study the closure properties of the family of languages accepted by [Formula: see text]-automata without [Formula: see text]-transitions. As a further consequence, when [Formula: see text] is the product of two groups, we give a characterization of the family of languages accepted by [Formula: see text]-automata in terms of the families of languages accepted by [Formula: see text]- and [Formula: see text]-automata. We also give a construction of a grammar for the language accepted by an arbitrary [Formula: see text]-automaton and show how to get a context-sensitive grammar when [Formula: see text] is finitely generated with a context-sensitive word problem and the [Formula: see text]-automaton is without [Formula: see text]-transitions.

Sign in / Sign up

Export Citation Format

Share Document