scholarly journals To the question of existence of exact irreducible representations of soluble groups of finite rank over locally finite field

2021 ◽  
Vol 15 ◽  
pp. 150
Author(s):  
A.V. Tushev
Keyword(s):  
Finite Field ◽  
Finite Rank ◽  
Sufficient Conditions ◽  
Irreducible Representations ◽  
Characteristic Subgroup ◽  
Locally Finite ◽  
Soluble Groups ◽  
Torsion Free Group ◽  
Locally Finite Field ◽  
Torsion Free

We find characteristic subgroup of soluble torsion-free group of finite rank, whose structure determines sufficient conditions of existence of exact irreducible representations of the group over locally finite field.

scholarly journals Spectra of conjugated ideals in group algebras of abelian groups of finite rank and control theorems

1996 ◽  
Vol 38 (3) ◽  
pp. 309-320 ◽  
Author(s):  
Anatolii V. Tushev
Keyword(s):  
Finite Rank ◽  
Torsion Group ◽  
Group Algebras ◽  
Prime Ideals ◽  
Locally Finite ◽  
Torsion Free Group ◽  
Free Rank ◽  
And Control ◽  
Torsion Free Rank ◽  
Torsion Free

Throughout kwill denote a field. If a group Γ acts on aset A we say an element is Γ-orbital if its orbit is finite and write ΔΓ(A) for the subset of such elements. Let I be anideal of a group algebra kA; we denote by I+ the normal subgrou(I+1)∩A of A. A subgroup B of an abelian torsion-free group A is said to be dense in A if A/B is a torsion-group. Let I be an ideal of a commutative ring K; then the spectrum Sp(I) of I is the set of all prime ideals P of K such that I≤P. If R is a ring, M is an R-module and x ɛ M we denote by the annihilator of x in R. We recall that a group Γ is said to have finite torsion-free rank if it has a finite series in which each factoris either infinite cyclic or locally finite; its torsion-free rank r0(Γ) is then defined to be the number of infinite cyclic factors in such a series.

scholarly journals ON THE IRREDUCIBLE REPRESENTATIONS OF SOLUBLE GROUPS OF FINITE RANK

2012 ◽  
Vol 05 (04) ◽  
pp. 1250061 ◽  
Author(s):  
A. V. Tushev
Keyword(s):  
Finite Rank ◽  
Soluble Group ◽  
Necessary Conditions ◽  
Irreducible Representations ◽  
Locally Finite ◽  
Soluble Groups ◽  
Sufficient And Necessary Conditions

We obtained some sufficient and necessary conditions of existence of faithful irreducible representations of a soluble group G of finite rank over a field k. It was shown that the existence of such representations strongly depends on construction of the socle of the group G. The situation is especially complicated in the case where the field k is locally finite.

scholarly journals Groups which satisfy a weak form of Poincaré duality

1991 ◽  
Vol 34 (3) ◽  
pp. 463-486
Author(s):  
J. E. Roberts
Keyword(s):  
Finite Rank ◽  
Weak Form ◽  
Homological Dimension ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Soluble Groups ◽  
Finite Dimensional ◽  
Duality Property ◽  
The Given ◽  
Torsion Free

Our main result is that a “restricted Poincaré duality” property with respect to finite dimensional coefficient modules over a field holds for a certain class of groups which includes all soluble groups of finite Hirsch length. This relies on a generalisation to the given class of a module construction by Stammbach; an extension of his result on homological dimension to these groups is given. We also generalise the well-known result that torsion-free soluble groups of finite rank are countable.

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).

scholarly journals Subgroups, of Chevalley groups over a locally finite field, defined by a family of additive subgroups

2017 ◽  
Vol 102 (5-6) ◽  
pp. 792-798
Author(s):  
V. A. Koibaev ◽  
S. K. Kuklina ◽  
A. O. Likhacheva ◽  
Ya. N. Nuzhin
Keyword(s):  
Finite Field ◽  
Chevalley Groups ◽  
Locally Finite ◽  
Locally Finite Field

Automorphisms of Finite Order of Soluble Groups of Finite Rank

2021 ◽  
Vol 58 (1) ◽  
pp. 19-31
Author(s):  
Bertram A. F. Wehrfritz
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Recent Work ◽  
Finite Order ◽  
Finite Rank ◽  
Soluble Groups ◽  
Torsion Free

We study the effect on sections of a soluble-by-finite group G of finite rank of an almost fixed-point-free automorphism φ of G of finite order. We also elucidate the structure of G if φ has order 4 and if G is also (torsion-free)-by-finite. The latter extends recent work of Xu, Zhou and Liu.

CONJUGATELY DENSE SUBGROUPS OF 3-DIMENSIONAL LINEAR GROUPS OVER LOCALLY FINITE FIELD

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1273-1280 ◽  
Author(s):  
S. A. ZYUBIN
Keyword(s):  
Finite Field ◽  
Linear Groups ◽  
Dense Subgroup ◽  
Locally Finite ◽  
3 Dimensional ◽  
Intermediate Group ◽  
Kourovka Notebook ◽  
Locally Finite Field ◽  
Conjugately Dense Subgroup ◽  
Dense Subgroups

A subgroup of any group is called conjugately dense if it has nonempty intersection with each class of conjugate elements of the group. The aim of this paper is to prove the following. Let K be a locally finite field and H be an irreducible conjugately dense subgroup of the intermediate group SL 3(K) ≤ G ≤ GL 3(K); then H = G. This result confirms part of P. Neumann's conjecture from problem 6.38 in "Kourovka Notebook" for the group GL 3(K) over locally finite field K.

scholarly journals Group algebras of some generalised soluble groups

1972 ◽  
Vol 18 (1) ◽  
pp. 1-5 ◽  
Author(s):  
R. P. Knott
Keyword(s):  
Free Group ◽  
Group Algebra ◽  
Soluble Group ◽  
Group Algebras ◽  
Soluble Groups ◽  
Torsion Free Group ◽  
Open Question ◽  
Torsion Free

In (8) Stonehewer referred to the following open question due to Amitsur: If G is a torsion-free group and F any field, is the group algebra, FG, of G over F semi-simple? Stonehewer showed the answer was in the affirmative if G is a soluble group. In this paper we show the answer is again in the affirmative if G belongs to a class of generalised soluble groups

scholarly journals Soluble groups isomorphic to their non-nilpotent subgroups

1999 ◽  
Vol 67 (3) ◽  
pp. 399-411 ◽  
Author(s):  
Howard Smith ◽  
James Wiegold
Keyword(s):  
Finite Rank ◽  
Free Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Torsion Free

AbstractA group G belongs to the class W if G has non-nilpotent proper subgroups and is isomorphic to all of them. The main objects of study are the soluble groups in W that are not finitely generated. It is proved that there are no torsion-free groups of this sort, and a reasonable classification is given in the finite rank case.

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