scholarly journals Decomposing finite Abelian groups

2001 ◽  
Vol 1 (3) ◽  
pp. 26-32
Author(s):  
K Cheung ◽  
M Mosca
Keyword(s):  
Efficient Algorithm ◽  
Riemann Hypothesis ◽  
Abelian Groups ◽  
Quantum Algorithm ◽  
Class Numbers ◽  
Finite Abelian Groups ◽  
Generalized Riemann Hypothesis ◽  
Cyclic Groups

This paper describes a quantum algorithm for efficiently decomposing finite Abelian groups into a product of cyclic groups. Such a decomposition is needed in order to apply the Abelian hidden subgroup algorithm. Such a decomposition (assuming the Generalized Riemann Hypothesis) also leads to an efficient algorithm for computing class numbers (known to be at least as difficult as factoring).

scholarly journals On Keller's conjecture for certain cyclic groups

1979 ◽  
Vol 22 (1) ◽  
pp. 17-21 ◽  
Author(s):  
A. D. Sands
Keyword(s):  
Prime Power ◽  
Abelian Groups ◽  
Prime Power Order ◽  
Finite Abelian Groups ◽  
Cyclic Groups

Keller (6) considered a generalisation of a problem of Minkowski (7) concerning the filling of Rn by congruent cubes. Hajós (4) reduced Minkowski's conjecture to a problem concerning the factorization of finite abelian groups and then solved this problem. In a similar manner Hajós (5) reduced Keller's conjecture to a problem in the factorization of finite abelian groups, but this problem remains unsolved, in general. It occurs also as Problem 80 in Fuchs (3). Seitz (10) has obtained a solution for cyclic groups of prime power order. In this paper we present a solution for cyclic groups whose order is the product of two prime powers.

scholarly journals The Number of Subgroup Chains of Finite Nilpotent Groups

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1537 ◽  
Author(s):  
Lingling Han ◽  
Xiuyun Guo
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Recursive Formula ◽  
Nilpotent Groups ◽  
Classification Problem ◽  
Finite Abelian Groups ◽  
Counting Problem ◽  
Cyclic Groups ◽  
Fuzzy Subgroups

In this paper, we mainly count the number of subgroup chains of a finite nilpotent group. We derive a recursive formula that reduces the counting problem to that of finite p-groups. As applications of our main result, the classification problem of distinct fuzzy subgroups of finite abelian groups is reduced to that of finite abelian p-groups. In particular, an explicit recursive formula for the number of distinct fuzzy subgroups of a finite abelian group whose Sylow subgroups are cyclic groups or elementary abelian groups is given.

scholarly journals Antiautomorphisms and Biantiautomorphisms of Some Finite Abelian Groups

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 294
Author(s):  
Daniel López-Aguayo ◽  
Servando López Aguayo
Keyword(s):  
Lower Bound ◽  
Abelian Groups ◽  
Additive Group ◽  
Exact Formula ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Partial Classification ◽  
Open Questions ◽  
Odd Order

We extend the concepts of antimorphism and antiautomorphism of the additive group of integers modulo n, given by Gaitanas Konstantinos, to abelian groups. We give a lower bound for the number of antiautomorphisms of cyclic groups of odd order and give an exact formula for the number of linear antiautomorphisms of cyclic groups of odd order. Finally, we give a partial classification of the finite abelian groups which admit antiautomorphisms and state some open questions.

scholarly journals UNCERTAINTY PRINCIPLES FOR GROUPS AND RECONSTRUCTION OF SIGNALS

2017 ◽  
Vol 21 (6) ◽  
pp. 102-109
Author(s):  
S.Y. Novikov ◽  
M.E. Fedina
Keyword(s):  
Abelian Groups ◽  
Summation Formula ◽  
Uncertainty Principles ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Finite Dimensional ◽  
Discrete Signals ◽  
Minimum Number ◽  
Dimensional Version ◽  
Indicator Functions

Uncertainty principles of harmonic analysis and their analogues for finite abelian groups are considered in the paper. Special attention is paid to the recent results of T. Tao and coauthors about cyclic groups of prime order. It is shown, that indicator functions of subgroups of finite Abelian groups are analogues of Gaussian functions. Finite-dimensional version of Poisson summation formula is proved. Opportunities of application of these results for reconstruction of discrete signals with incomplete number of coefficients are suggested. The principle of partial isometric whereby we can determine the minimum number of measurements for stable recovery of the signal are formulated.

scholarly journals Comments on Y. O. Hamidoune's Paper ‘Adding Distinct Congruence Classes’

2016 ◽  
Vol 25 (5) ◽  
pp. 641-644
Author(s):  
BÉLA BAJNOK
Keyword(s):  
Cyclic Group ◽  
Abelian Groups ◽  
Short Note ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Even Order ◽  
Congruence Classes

The main result in Y. O. Hamidoune's paper ‘Adding distinct congruence classes' (Combin. Probab. Comput.7 (1998) 81–87) is as follows. If S is a generating subset of a cyclic group G such that 0 ∉ S and |S| ⩾ 5, then the number of sums of the subsets of S is at least min(|G|, 2|S|). Unfortunately, the argument of the author, who, sadly, passed away in 2011, relies on a lemma whose proof is incorrect; in fact, the lemma is false for all cyclic groups of even order. In this short note we point out this mistake, correct the proof, and discuss why the main result is actually true for all finite abelian groups.

scholarly journals Computational methods for difference families in finite abelian groups

2019 ◽  
Vol 7 (1) ◽  
pp. 127-141
Author(s):  
Dragomir Ž. Ðoković ◽  
Ilias S. Kotsireas
Keyword(s):  
Spectral Density ◽  
Power Spectral Density ◽  
Computational Methods ◽  
Abelian Groups ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Difference Families ◽  
Power Spectral

Abstract 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 abelian groups. In particular the power spectral density test and the method of compression can be used to help the search.

scholarly journals On finite loops whose inner mapping groups are Abelian II

2005 ◽  
Vol 71 (3) ◽  
pp. 487-492
Author(s):  
Markku Niemenmaa
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Prime Power ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Prime Power Order ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Inner Mapping Group ◽  
Mapping Group

If the inner mapping group of a loop is a finite Abelian group, then the loop is centrally nilpotent. We first investigate the structure of those finite Abelian groups which are not isomorphic to inner mapping groups of loops and after this we show that if the inner mapping group of a loop is isomorphic to the direct product of two cyclic groups of the same odd prime power order pn, then our loop is centrally nilpotent of class at most n + 1.

scholarly journals Isomorphisms of Direct Products of Finite Cyclic Groups

2012 ◽  
Vol 20 (4) ◽  
pp. 343-347
Author(s):  
Kenichi Arai ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Fundamental Theorem ◽  
Chinese Remainder Theorem ◽  
Abelian Groups ◽  
Finite Sequence ◽  
Direct Products ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Relative Prime

Summary In this article, we formalize that every finite cyclic group is isomorphic to a direct product of finite cyclic groups which orders are relative prime. This theorem is closely related to the Chinese Remainder theorem ([18]) and is a useful lemma to prove the basis theorem for finite abelian groups and the fundamental theorem of finite abelian groups. Moreover, we formalize some facts about the product of a finite sequence of abelian groups.

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