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.

scholarly journals Direct Sum Cancellation of Noetherian Modules

1998 ◽  
Vol 200 (1) ◽  
pp. 207-224 ◽  
Author(s):  
Gary Brookfield
Keyword(s):  
Direct Sum ◽  
Noetherian Modules

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.

On the structure of some Noetherian modules over group rings

2014 ◽  
Vol 24 (02) ◽  
pp. 233-249
Author(s):  
Leonid A. Kurdachenko ◽  
Javier Otal ◽  
Igor Ya. Subbotin
Keyword(s):  
Finite Field ◽  
Group Ring ◽  
Soluble Group ◽  
Dedekind Domain ◽  
Group Rings ◽  
Property A ◽  
Locally Finite ◽  
Locally Finite Field ◽  
Noetherian Modules

In this paper, we study the structure of some Noetherian modules over group rings and deduce some statements regarding the structure of the groups involved. More precisely, we consider a module A over a group ring RG with the following property: A is a Noetherian RH-module for every subgroup H, which is not contained in the centralizer CG(A). If G is some generalized soluble group and R is a locally finite field or some Dedekind domain, we describe the structure of G/CG(A).

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.

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.

Structure of Some Noetherian Injective Modules

1980 ◽  
Vol 32 (6) ◽  
pp. 1277-1287 ◽  
Author(s):  
B. Sarath
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Endomorphism Rings ◽  
Commutative Case ◽  
Noetherian Module ◽  
Direct Sum ◽  
Injective Modules ◽  
Semiprime Rings ◽  
Noetherian Modules

The main object of this paper is to study when infective noetherian modules are artinian. This question was first raised by J. Fisher and an example of an injective noetherian module which is not artinian is given in [9]. However, it is shown in [4] that over commutative rings, and over hereditary noetherian P.I rings, injective noetherian does imply artinian. By combining results of [6] and [4] it can be shown that the above implication is true over any noetherian P.I ring. It is shown in this paper that injective noetherian modules are artinian over rings finitely generated as modules over their centers, and over semiprime rings of Krull dimension 1. It is also shown that every injective noetherian module over a P.I ring contains a simple submodule. Since any noetherian injective module is a finite direct sum of indecomposable injectives it suffices to study when such injectives are artinian. IfQis an injective indecomposable noetherian module, thenQcontains a non-zero submoduleQ0such that the endomorphism rings ofQ0and all its submodules are skewfields. Over a commutative ring, such aQ0is simple. In the non-commutative case very little can be concluded, and many of the difficulties seem to arise here.

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.

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