scholarly journals GROUPS WITH MANY PRONORMAL SUBGROUPS

2021 ◽  
pp. 1-12
Author(s):  
MARIA FERRARA ◽  
MARCO TROMBETTI
Keyword(s):  
Soluble Group ◽  
Regular Cardinality ◽  
Locally Soluble Group

Abstract A subgroup H of a group G is pronormal in G if each of its conjugates $H^g$ in G is conjugate to it in the subgroup $\langle H,H^g\rangle $ ; a group is prohamiltonian if all of its nonabelian subgroups are pronormal. The aim of the paper is to show that a locally soluble group of (regular) cardinality in which all proper uncountable subgroups are prohamiltonian is prohamiltonian. In order to obtain this result, it is proved that the class of prohamiltonian groups is detectable from the behaviour of countable subgroups. Examples are exhibited to show that there are uncountable prohamiltonian groups that do not behave very well. Finally, it is shown that prohamiltonicity can sometimes be detected through the analysis of the finite homomorphic images of a group.

scholarly journals A group variety defined by a semigroup law

1996 ◽  
Vol 60 (2) ◽  
pp. 255-259 ◽  
Author(s):  
A. Yu. Ol'shanskii ◽  
A. Storozhev
Keyword(s):  
Free Group ◽  
Soluble Group ◽  
Group Variety ◽  
Locally Soluble Group

AbstractA group variety defined by one semigroup law in two variables is constructed and it is proved that its free group is not a periodic extension of a locally soluble group.

The structure of noetherian modules over hyperfinite groups

1992 ◽  
Vol 112 (1) ◽  
pp. 21-28 ◽  
Author(s):  
Z. Y. Duan
Keyword(s):  
Soluble Group ◽  
Direct Sum ◽  
Locally Soluble Group ◽  
Noetherian Modules

Let G be a hyperfinite locally soluble group and let A be a noetherian ℤZ;G-module. In [2], we proved that A is the direct sum of a ℤZ;G-submodule Af each of whose irreducible ℤZ;G-module sections is finite and a ℤZ;G-submodule each of whose irreducible ℤZ;G-module sections is infinite. In this paper we study the structure of the ℤZ;G-submodules Af and . Our main result gives a complete description of Af.

ON MODULES OVER GROUP RINGS OF LOCALLY SOLUBLE GROUPS WITH RANK RESTRICTIONS ON SOME SYSTEMS OF SUBGROUPS

2010 ◽  
Vol 03 (01) ◽  
pp. 45-55 ◽  
Author(s):  
O. Yu. Dashkova
Keyword(s):  
Soluble Group ◽  
Proper Subgroup ◽  
Dedekind Domain ◽  
Group Rings ◽  
Soluble Groups ◽  
Rank Restrictions ◽  
Locally Soluble Group ◽  
Dg Module

We consider a DG-module A over a Dedekind domain D. Let G be a group having infinite section p-rank (or infinite 0-rank) such that CG(A) = 1. It is known that if A is not an artinian D-module then for every proper subgroup H, the quotient A/CA(H) is an artinian D-module for every proper subgroup H of infinite section p-rank (or infinite 0-rank respectively). In this paper, it is proved that if G is a locally soluble group, then G is soluble. Some properties of soluble groups of this type are also obtained.

scholarly journals The decomposition of artinian modules over hyper-(cyclic or finite) groups

1996 ◽  
Vol 39 (1) ◽  
pp. 115-118
Author(s):  
Y. B. Qin
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Artinian Modules ◽  
Locally Soluble Group

If G is a hyperfinite locally soluble group and A an artinian ZG-module then Zaĭcev proved that A has an f-decomposition. For G being a hyper-(cyclic or finite) locally soluble group, Z. Y. Duan has shown that any periodic artinian ZG-module A has an f-decomposition. Here we prove that: if G is a hyper-(cyclic or finite) group, then any artinian ZG-module A has an f-decomposition.

Groups with few conjugacy classes of non-normal subgroups

2008 ◽  
Vol 58 (2) ◽  
Author(s):  
Maria Falco ◽  
Francesco Giovanni ◽  
Carmela Musella
Keyword(s):  
Abelian Subgroup ◽  
Soluble Group ◽  
Commutator Subgroup ◽  
Conjugacy Classes ◽  
Normal Subgroups ◽  
Abelian Subgroups ◽  
Locally Soluble Group

AbstractThe structure of groups with finitely many non-normal subgroups is well known. In this paper, groups are investigated with finitely many conjugacy classes of non-normal subgroups with a given property. In particular, it is proved that a locally soluble group with finitely many non-trivial conjugacy classes of non-abelian subgroups has finite commutator subgroup. This result generalizes a theorem by Romalis and Sesekin on groups in which every non-abelian subgroup is normal.

On some groups with only two types of subgroups

2014 ◽  
Vol 07 (04) ◽  
pp. 1450057
Author(s):  
L. A. Kurdachenko ◽  
I. Ya. Subbotin ◽  
T. I. Velichko
Keyword(s):  
Soluble Group ◽  
Soluble Groups ◽  
Locally Soluble Group

In this paper, we study the groups whose subgroups are either almost normal or contranormal. The main theorem of the paper describes locally soluble groups with this property. We also provide a description of locally soluble group, whose subgroups are almost normal or abnormal.

scholarly journals ON SOLUBILITY OF GROUPS WITH BOUNDED CENTRALIZER CHAINS

2009 ◽  
Vol 51 (1) ◽  
pp. 49-54 ◽  
Author(s):  
E. I. KHUKHRO
Keyword(s):  
Soluble Group ◽  
Maximum Length ◽  
Finite Soluble Group ◽  
Derived Length ◽  
A Chain ◽  
Locally Soluble Group

AbstractThe c-dimension of a group is the maximum length of a chain of nested centralizers. It is proved that a periodic locally soluble group of finite c-dimension k is soluble of derived length bounded in terms of k, and the rank of its quotient by the Hirsch–Plotkin radical is bounded in terms of k. Corollary: a pseudo-(finite soluble) group of finite c-dimension k is soluble of derived length bounded in terms of k.

scholarly journals The stable subgroup graph

2018 ◽  
Vol 36 (3) ◽  
pp. 129-139
Author(s):  
Behnaz Tolue
Keyword(s):  
Abelian Group ◽  
Soluble Group ◽  
Normal Subgroups ◽  
Induced Subgraph ◽  
Vertex Set

In this paper we introduce stable subgroup graph associated to the group $G$. It is a graph with vertex set all subgroups of $G$ and two distinct subgroups $H_1$ and $H_2$ are adjacent if $St_{G}(H_1)\cap H_2\neq 1$ or $St_{G}(H_2)\cap H_1\neq 1$. Its planarity is discussed whenever $G$ is an abelian group, $p$-group, nilpotent, supersoluble or soluble group. Finally, the induced subgraph of stable subgroup graph with vertex set whole non-normal subgroups is considered and its planarity is verified for some certain 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.

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