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COMPUTABLE ABELIAN GROUPS

ALEXANDER G. MELNIKOV
2014 Bulletin of Symbolic Logic  
We provide an introduction to methods and recent results on infinitely generated abelian groups with decidable word problem.  ...  A computable ordered abelian group is a computable abelian group with a computable order on it.  ...  As we will see, computably categorical abelian groups fall into this general pattern. Thus, we typically have to deal with computable abelian groups that are not computably categorical.  ... 
doi:10.1017/bsl.2014.32 fatcat:utyr6wtenjh5bg7vm4kkinsocu

Computable topological abelian groups [article]

Martino Lupini, Alexander Melnikov, Andre Nies
2021 arXiv   pre-print
We prove that the dualization is computable in the important cases of compact and locally compact totally disconnected Polish abelian groups.  ...  among all Polish groups.  ...  -presented torsion-free abelian group A has a computable presentation.  ... 
arXiv:2105.12897v2 fatcat:zgeiw3nxdrd2nhmam4qy5haj5e

Computers, trees and Abelian groups

F. Richman
1988 Computers and Mathematics with Applications  
The lattices of trees that correspond to indecomposable pairs that are bounded by pn were generated by computer up to n = 6.  ...  As is the case for finite Abelian groups themselves, this problem can easily be reduced to the study of p-groups --groups in which each element has order a power of p, for some fixed prime p.  ...  There seems to be no natural connection of a tree with its dual, nor could we find some grander duality in Abelian group theory of which this duality is a reflection.  ... 
doi:10.1016/0898-1221(88)90220-9 fatcat:6ihtucttcrbz7mayd34q563rum

Computable Ordered Abelian Groups and Fields [chapter]

Alexander G. Melnikov
2010 Lecture Notes in Computer Science  
We present transformations of linearly ordered sets into ordered abelian groups and ordered fields. We study effective properties of the transformations.  ...  In particular, we show that a linear order L has a ∆ 0 2 copy if and only if the corresponding ordered group (ordered field) has a computable copy.  ...  We study complexity of isomorphisms between computable copies of ordered abelian groups and fields 1 .  ... 
doi:10.1007/978-3-642-13962-8_36 fatcat:g4l3m5slonhthpcw7chpviqegy

Computing in general abelian groups is hard

Costas S. Iliopoulos
1985 Theoretical Computer Science  
an element, (iii) membership testing, (iv) testing whether or not a group is cyclic, (v) computing the canonical structure of an abelian group.  ...  The relative complexity of the following problems on abelian groups represented by an explicit set of generators is investigated: (i) computing a set of defining relations, (ii) computing the order of  ...  Given abelian groups G= (S) and F= (S'), with IGI n and IFI m, and the fact that (Fu G) is abelian, compute the order of (Fr~ G).  ... 
doi:10.1016/0304-3975(85)90061-1 fatcat:m653sjzcmre35j7wl652h6a4tq

The computable dimension of ordered abelian groups

Sergey S. Goncharov, Steffen Lempp, Reed Solomon
2003 Advances in Mathematics  
Let G be a computable ordered abelian group.  ...  We show that the computable dimension of G is either 1 or o; that G is computably categorical if and only if it has finite rank, and that if G has only finitely many Archimedean classes, then G has a computable  ...  The triple ðG; þ G ; p G Þ is called a computable ordered abelian group.  ... 
doi:10.1016/s0001-8708(02)00042-7 fatcat:34t4gfjw75fyhf6ioqrsbtozk4

Computation of invariants of finite abelian groups

Evelyne Hubert, George Labahn
2016 Mathematics of Computation  
We investigate the computation and applications of rational invariants of the linear action of a finite abelian group in the non-modular case.  ...  As an application, we provide a symmetry reduction scheme for polynomial systems whose solution set is invariant by a finite abelian group action.  ...  Introduction Recently Faugère and Svartz [7] demonstrated how to reduce the complexity of Gröbner bases computations for ideals stable by the linear action of a finite abelian group in the non modular  ... 
doi:10.1090/mcom/3076 fatcat:5imz6omhwfeirap54nkgxmtcci

Structure computation and discrete logarithms in finite abelian $p$-groups

Andrew V. Sutherland
2010 Mathematics of Computation  
We present a generic algorithm for computing discrete logarithms in a finite abelian p-group H, improving the Pohlig-Hellman algorithm and its generalization to noncyclic groups by Teske.  ...  The problem of computing a basis for some or all of the Sylow p-subgroups of an arbitrary finite abelian group G is addressed, yielding a Monte Carlo algorithm to compute the structure of G using O(|G|  ...  Discrete logarithms may be applied to compute the structure of a finite abelian group.  ... 
doi:10.1090/s0025-5718-10-02356-2 fatcat:6f4iuafaunbzzi52p34bjxh6rq

Computing the structure of a finite abelian group

Johannes Buchmann, Arthur Schmidt
2005 Mathematics of Computation  
We present an algorithm that computes the structure of a finite abelian group G from a generating system M . The algorithm executes O(|M | |G|) group operations and stores O( |G|) group elements.  ...  Introduction Let G be a finite abelian group.  ...  Computing the structure of the finite abelian group G from the gen- erating system M requires storing O( |G|) pairs (g, q) ∈ G × {0, . . . , |G| } |M | , O(|M | |G|) multiplications and inversions in G  ... 
doi:10.1090/s0025-5718-05-01740-0 fatcat:wwctvgp6kzcwjkhmwm73b53v7q

On some computational problems in finite abelian groups

Johannes Buchmann, Michael J. Jacobson Jr., Edlyn Teske
1997 Mathematics of Computation  
We present new algorithms for computing orders of elements, discrete logarithms, and structures of finite abelian groups.  ...  discrete logarithm, or size of the group, rather than relative to an upper bound on the group order.  ...  This algorithm computes the HNF-basis for the lattice of relations on a generating system for a finite abelian group.  ... 
doi:10.1090/s0025-5718-97-00880-6 fatcat:ueciysoamrfltn5ehs7kdnxebq

k-Sums in Abelian Groups

BENJAMIN GIRARD, SIMON GRIFFITHS, YAHYA OULD HAMIDOUNE
2012 Combinatorics, probability & computing  
Given a finite subset A of an abelian group G, we study the set k ∧ A of all sums of k distinct elements of A. In this paper, we prove that |k ∧ A| >= |A| for all k in 2,...  ...  Many results concerning k-sums of subsets of abelian groups focus on the cases k = 2 and k = 3.  ...  For example, one might hope to prove an analogue of (1) for additive subsets A of general abelian groups.  ... 
doi:10.1017/s0963548312000168 fatcat:3ymk6axrnfeqtj3voymoholznu

Decidability and Computability of Certain Torsion-Free Abelian Groups

Rodney G. Downey, Sergei S. Goncharov, Asher M. Kach, Julia F. Knight, Oleg V. Kudinov, Alexander G. Melnikov, Daniel Turetsky
2010 Notre Dame Journal of Formal Logic  
We study completely decomposable torsion-free abelian groups of the form G S := ⊕ n∈S Qp n for sets S ⊆ ω.  ...  We show that G S has a decidable copy if and only if S is Σ 0 2 and has a computable copy if and only if S is Σ 0 3 .  ...  A completely decomposable torsion-free abelian group A ∼ = i∈I A i is effectively (strongly) decomposable if it has a computable (decidable) copy in which the predicates P i (x) x ∈ A i are uniformly computable  ... 
doi:10.1215/00294527-2010-006 fatcat:nivejewytzanxak4ki4awl4pma

Some computations of non-abelian tensor products of groups

R Brown, D.L Johnson, E.F Robertson
1987 Journal of Algebra  
Given an abelian group A, TA is the abelian group with generators ya, aEA, and defining relations Ha-'1 = 74 y(abc) ya yb YC = dab) y(bc) y(caX (12) for all a, 6, c E A.  ...  The following result was suggested by the computational results of Section 6. PROPOSITION 8. Zf G is a group in which G' has a cyclic complement C, then G@Gr(G A G)xC.  ...  Use the ideas in the proof of Proposition 8 to compute the tensor square of GL(2, p) and other linear groups. (1) .  ... 
doi:10.1016/0021-8693(87)90248-1 fatcat:mcqbnetffngylc2n6ggdlasksu

Computational methods for difference families in finite abelian groups

Dragomir Ž. Ðoković, Ilias S. Kotsireas
2019 Special Matrices  
abelian groups.  ...  Our main objective is to show that the computational methods, developed previously to search for difference families in cyclic groups, can be fully extended to the more general case of arbitrary finite  ...  As an example let us compute the DFT of the cyclic group G = g of order v with a generator g.  ... 
doi:10.1515/spma-2019-0012 fatcat:drbwkfulm5ddhpqypp3vphsuty

Computing generators of the unit group of an integral abelian group ring [article]

Paolo Faccin, Willem A. de Graaf, Wilhelm Plesken
2013 arXiv   pre-print
We describe an algorithm for obtaining generators of the unit group of the integral group ring ZG of a finite abelian group G.  ...  We used our implementation in Magma of this algorithm to compute the unit groups of ZG for G of order up to 110.  ...  We have computed the Hoechsmann indices for all abelian groups of orders up to 110. For most groups the index is 1.  ... 
arXiv:1301.1770v1 fatcat:z6i36xaxizd55bjnepkct64x3q
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