Classification theory of abelian groups, II: Local theory

1981 ◽  
pp. 322-349 ◽  
Author(s):  
Robert B. Warfield
Keyword(s):  
Abelian Groups ◽  
Classification Theory ◽  
Local Theory

scholarly journals Varieties and finite groups

1969 ◽  
Vol 10 (1-2) ◽  
pp. 5-19 ◽  
Author(s):  
L. G. Kovács
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Classification Theory ◽  
Master Plan ◽  
Initial Part ◽  
Finitely Generated ◽  
The Future ◽  
Historical Accident ◽  
Special Classes

The theory of discrete, abstract groups, as presented in current texts, consists of investigations of various special classes of groups: it has very few completely general results. For some classes (say, for finite groups) the investigations have been extensive and successful; in a few cases (say, for finitely generated abelian groups) they have even reached a sense of completeness. The choice of some of these classes was dictated by the needs of other branches of mathematics. Many more were introduced with the view of extending the scope of certain powerful but special results, and a large part of the literature is taken up by elaborate counterexamples which mark the limits of these generalizations. In so far as one is looking for some kind of classification theory, it is immediately evident that the classes investigated were chosen by historical accident rather than by any master plan, and so far do not appear to form the initial part of a pattern which could be enlarged and completed in the future.

Classification theory for abelian groups with an endomorphism

1991 ◽  
Vol 31 (2) ◽  
pp. 95-104 ◽  
Author(s):  
Annalisa Marcja ◽  
Mike Prest ◽  
Carlo Toffalori
Keyword(s):  
Abelian Groups ◽  
Classification Theory

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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