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 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.

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.

Adding Distinct Congruence Classes

1998 ◽  
Vol 7 (1) ◽  
pp. 81-87 ◽  
Author(s):  
Y. O. HAMIDOUNE
Keyword(s):  
Cyclic Group ◽  
Abelian Groups ◽  
Congruence Classes ◽  
Nonabelian Groups

Let S be a generating subset of a cyclic group G such that 0=∉S and [mid ]S[mid ][ges ]5. We show that the number of sums of the subsets of S is at least min([mid ]G[mid ], 2[mid ]S[mid ]). Our bound is best possible. We obtain similar results for abelian groups and mention the generalization to nonabelian groups.

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 EXTENSION OF AUTOMORPHISMS OF SUBGROUPS

2012 ◽  
Vol 54 (2) ◽  
pp. 371-380
Author(s):  
G. G. BASTOS ◽  
E. JESPERS ◽  
S. O. JURIAANS ◽  
A. DE A. E SILVA
Keyword(s):  
Abelian Group ◽  
Finite Groups ◽  
Dihedral Group ◽  
Abelian Groups ◽  
Alternating Group ◽  
Finite Abelian Groups ◽  
Quaternion Group ◽  
Icosahedral Group ◽  
Odd Order ◽  
Even Order

AbstractLet G be a group such that, for any subgroup H of G, every automorphism of H can be extended to an automorphism of G. Such a group G is said to be of injective type. The finite abelian groups of injective type are precisely the quasi-injective groups. We prove that a finite non-abelian group G of injective type has even order. If, furthermore, G is also quasi-injective, then we prove that G = K × B, with B a quasi-injective abelian group of odd order and either K = Q8 (the quaternion group of order 8) or K = Dih(A), a dihedral group on a quasi-injective abelian group A of odd order coprime with the order of B. We give a description of the supersoluble finite groups of injective type whose Sylow 2-subgroup are abelian showing that these groups are, in general, not quasi-injective. In particular, the characterisation of such groups is reduced to that of finite 2-groups that are of injective type. We give several restrictions on the latter. We also show that the alternating group A5 is of injective type but that the binary icosahedral group SL(2, 5) is not.

scholarly journals Identity Graph of Finite Cyclic Groups

2021 ◽  
Vol 03 (01) ◽  
pp. 101-110
Author(s):  
Maria Vianney Any Herawati ◽  
◽  
Priscila Septinina Henryanti ◽  
Ricky Aditya ◽  
◽  
...  
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Identity Element ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Even Order ◽  
A Finite Group

This paper discusses how to express a finite group as a graph, specifically about the identity graph of a cyclic group. The term chosen for the graph is an identity graph, because it is the identity element of the group that holds the key in forming the identity graph. Through the identity graph, it can be seen which elements are inverse of themselves and other properties of the group. We will look for the characteristics of identity graph of the finite cyclic group, for both cases of odd and even order.

scholarly journals ON THE MAXIMUM SIZE OF A (k,l)-SUM-FREE SUBSET OF AN ABELIAN GROUP

2009 ◽  
Vol 05 (06) ◽  
pp. 953-971 ◽  
Author(s):  
BÉLA BAJNOK
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Maximum Size ◽  
Finite Abelian Groups ◽  
Free Subset ◽  
Free Set ◽  
Free Sets

A subset A of a given finite abelian group G is called (k,l)-sum-free if the sum of k (not necessarily distinct) elements of A does not equal the sum of l (not necessarily distinct) elements of A. We are interested in finding the maximum size λk,l(G) of a (k,l)-sum-free subset in G. A (2,1)-sum-free set is simply called a sum-free set. The maximum size of a sum-free set in the cyclic group ℤn was found almost 40 years ago by Diamanda and Yap; the general case for arbitrary finite abelian groups was recently settled by Green and Ruzsa. Here we find the value of λ3,1(ℤn). More generally, a recent paper by Hamidoune and Plagne examines (k,l)-sum-free sets in G when k - l and the order of G are relatively prime; we extend their results to see what happens without this assumption.

CHARACTERIZATIONS OF SOME WREATH PRODUCTS OF GROUPS BY THEIR ENDOMORPHISM SEMIGROUPS

2008 ◽  
Vol 18 (02) ◽  
pp. 243-255 ◽  
Author(s):  
PEETER PUUSEMP
Keyword(s):  
Abelian Group ◽  
Wreath Product ◽  
Cyclic Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Wreath Products ◽  
Endomorphism Semigroup ◽  
Finite Abelian Groups ◽  
Products Of Groups

Let A be a cyclic group of order pn, where p is a prime, and B be a finite abelian group or a finite p-group which is determined by its endomorphism semigroup in the class of all groups. It is proved that under these assumptions the wreath product A Wr B is determined by its endomorphism semigroup in the class of all groups. It is deduced from this result that if A, B, A0,…, An are finite abelian groups and A0,…, An are p-groups, p prime, then the wreath products A Wr B and An Wr (…( Wr (A1 Wr A0))…) are determined by their endomorphism semigroups in the class of all groups.

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