Artinian modules over abelian groups of finite section rank

2006 ◽  
pp. 167-177
Keyword(s):  
Abelian Groups ◽  
Artinian Modules ◽  
Finite Section ◽  
Section Rank

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.

scholarly journals On minimal artinian modules and minimal artinian linear groups

2001 ◽  
Vol 27 (12) ◽  
pp. 707-714
Author(s):  
Leonid A. Kurdachenko ◽  
Igor Ya. Subbotin
Keyword(s):  
Abelian Groups ◽  
Linear Groups ◽  
Artinian Modules

The paper is devoted to the study of some important types of minimal artinian linear groups. The authors prove that in such classes of groups as hypercentral groups (so also, nilpotent and abelian groups) andFC-groups, minimal artinian linear groups have precisely the same structure as the corresponding irreducible linear groups.

GROUPS IN WHICH THE NORMAL CLOSURES OF CYCLIC SUBGROUPS HAVE FINITE SECTION RANK

2012 ◽  
Vol 22 (04) ◽  
pp. 1250032
Author(s):  
LEONID A. KURDACHENKO ◽  
JOSÉ M. MUÑOZ-ESCOLANO ◽  
JAVIER OTAL
Keyword(s):  
Explicit Formulas ◽  
Cyclic Subgroups ◽  
Radical Group ◽  
Finite Section ◽  
Section Rank

A group G is said to have finite section p-rankrp(G) = r (here p is a prime) if every elementary abelian p-section U/V of G is finite of order at most pr and there is an elementary abelian p-section A/B of G such that |A/B| = pr. If ℙ is the set of all primes and λ : ℙ → ℕ ∪ {0} is a function, we say that a group G has λ-bounded section rank if rp(G) ≤ λ(p) for each p ∈ ℙ. In this paper we show that if G is a locally generalized radical group in which the normal closures of the cyclic subgroups of G have finite λ-bounded section rank, then [G,G] has [Formula: see text]-bounded section rank for some function [Formula: see text]. This is a wide generalization of some results by Neumann, Smith and many others. Moreover we are able to give explicit formulas for the involved bounds.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

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