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.

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 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.

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.

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 Subgroups like Wielandt's in soluble groups

2000 ◽  
Vol 42 (1) ◽  
pp. 67-74 ◽  
Author(s):  
Clara Franchi
Keyword(s):  
Soluble Group ◽  
Subject Classification ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
Subnormal Subgroups

For each m≥1, u_{m}(G) is defined to be the intersection of the normalizers of all the subnormal subgroups of defect at most m in G. An ascending chain of subgroups u_{m,i}(G) is defined by setting u_{m,i}(G)/u_{m,i−1}(G)=u_{m}(G/u_{m,i−1}(G)). If u_{m,n}(G)=G, for some integer n, the m-Wielandt length of G is the minimal of such n.In [3], Bryce examined the structure of a finite soluble group with given m-Wielandt length, in terms of its polynilpotent structure. In this paper we extend his results to infinite soluble groups.1991 Mathematics Subject Classification. 20E15, 20F22.

scholarly journals On complemented chief factors of finite soluble groups

1972 ◽  
Vol 7 (1) ◽  
pp. 101-104 ◽  
Author(s):  
D.W. Barnes
Keyword(s):  
Soluble Group ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
Chief Factors ◽  
One To One

Let G = H0 > H1 > … > Hr = 1 and G = K0 > K1 > … > Kr =1 be two chief series of the finite soluble group G. Suppose Mi complements Hi/Hi+1. Then Mi also complements precisely one factor Kj/Kj+1, of the second series, and this Kj/Kj+1 is G-isomorphic to Hi/Hi+1. It is shown that complements Mi can be chosen for the complemented factors Hi/Hi+1 of the first series in such a way that distinct Mi complement distinct factors of the second series, thus establishing a one-to-one correspondence between the complemented factors of the two series. It is also shown that there is a one-to-one correspondence between the factors of the two series (but not in general constructible in the above manner), such that corresponding factors are G-isomorphic and have the same number of complements.

Finite soluble groups admitting an automorphism of prime power order with few fixed points

1987 ◽  
Vol 102 (3) ◽  
pp. 431-441 ◽  
Author(s):  
Brian Hartley ◽  
Volker Turau
Keyword(s):  
Fixed Points ◽  
Soluble Group ◽  
Prime Power ◽  
Prime Power Order ◽  
Finite Soluble Group ◽  
Prime Divisors ◽  
Fitting Subgroup ◽  
Soluble Groups ◽  
Fitting Height ◽  
Number Of Prime Divisors

Let G be a finite soluble group with Fitting subgroup F(G). The Fitting series of G is defined, as usual, by F0(G) = 1 and Fi(G)/Fi−1(G) = F(G/Fi−1(G)) for i ≥ 1, and the Fitting height h = h(G) of G is the least integer such that Fn(G) = G. Suppose now that a finite soluble group A acts on G. Let k be the composition length of A, that is, the number of prime divisors (counting multiplicities) of |A|. There is a certain amount of evidence in favour of theCONJECTURE. |G:Fk(G)| is bounded by a number depending only on |A| and |CG(A)|.

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.

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.

scholarly journals Cover-Avoidance Properties in Finite Soluble Groups

1976 ◽  
Vol 19 (2) ◽  
pp. 213-216 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Soluble Group ◽  
Chief Factor ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
General Method

AbstractWe give a general method for constructing subgroups which either cover or avoid each chief factor of the finite soluble group G. A strongly pronorrnal subgroup V, a prefrattini subgroup W, an -normalizer D and intersections and products of V, W, and D axe all constructable. The constructable subgroups can be characterized by their cover-avoidance property and a permutability condition as in the results of J. D. Gillam [4] for prefrattini subgroups and -normalizers.

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