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 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 Zero-sum subsets of decomposable sets in Abelian groups

2020 ◽  
Vol 30 (1) ◽  
pp. 15-25
Author(s):  
T. Banakh ◽  
◽  
A. Ravsky ◽  
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Open Problem ◽  
Abelian Groups ◽  
The Other ◽  
Other Hand ◽  
Decomposable Sets ◽  
Empty Subset ◽  
Zero Sum

A subset D of an abelian group is decomposable if ∅≠D⊂D+D. In the paper we give partial answers to an open problem asking whether every finite decomposable subset D of an abelian group contains a non-empty subset Z⊂D with ∑Z=0. For every n∈N we present a decomposable subset D of cardinality |D|=n in the cyclic group of order 2n−1 such that ∑D=0, but ∑T≠0 for any proper non-empty subset T⊂D. On the other hand, we prove that every decomposable subset D⊂R of cardinality |D|≤7 contains a non-empty subset T⊂D of cardinality |Z|≤12|D| with ∑Z=0. For every n∈N we present a subset D⊂Z of cardinality |D|=2n such that ∑Z=0 for some subset Z⊂D of cardinality |Z|=n and ∑T≠0 for any non-empty subset T⊂D of cardinality |T|<n=12|D|. Also we prove that every finite decomposable subset D of an Abelian group contains two non-empty subsets A,B such that ∑A+∑B=0.

scholarly journals On lacunary sets for nonabelian groups

1978 ◽  
Vol 25 (2) ◽  
pp. 167-176 ◽  
Author(s):  
A. H. Dooley
Keyword(s):  
Necessary Conditions ◽  
Abelian Groups ◽  
Sidon Sets ◽  
Compact Abelian Groups ◽  
Nonabelian Groups

AbstractResults concerning a class of lacunary sets are generalized from compact abelian to compact nonabelian groups. This class was introduced for compact abelian groups by Bozejko and Pytlik; it includes the p-Sidon sets of Edwards and Ross. A notion of test family is introduced and is used to give necessary conditions for a set to be lacunary. Using this, it is shown that (2) has no infinite p-Sidon sets for 1 ≤p<2.

scholarly journals Finite metacyclic groups acting on bordered surfaces

1994 ◽  
Vol 36 (2) ◽  
pp. 233-240 ◽  
Author(s):  
Coy L. May
Keyword(s):  
Cyclic Group ◽  
Quotient Group ◽  
Commutator Subgroup ◽  
Cyclic Extension ◽  
Metacyclic Group ◽  
Dihedral Groups ◽  
Klein Surfaces ◽  
Metacyclic Groups ◽  
Nonabelian Groups

A group is called metacyclic in case both its commutator subgroup and commutator quotient group are cyclic. Thus a metacyclic group is a cyclic extension of a cyclic group, and metacyclic groups are among the best understood of the nonabelian groups. Many interesting groups are metacyclic. For instance, the dihedral groups and the “odd” dicyclic groups are metacyclic; see [4, pp. 9–11] for more examples. Here we shall consider the actions of these groups on bordered Klein surfaces.

scholarly journals Adders for the patterned starter in nonabelian groups

1976 ◽  
Vol 21 (2) ◽  
pp. 185-193 ◽  
Author(s):  
Kenneth B. Gross ◽  
Philip A. Leonard
Keyword(s):  
Large Class ◽  
Abelian Groups ◽  
Odd Order ◽  
Room Squares ◽  
Combinatorial Constructions ◽  
Nonabelian Groups

AbstractStarters with adders, in abelian groups of odd order, have been used widely in recent combinatorial constructions, most notably in the study of Room squares. In this paper, constructions of adders for the so-called “patterned starter” are described, for a large class of nonabelian groups of odd order.

Isomorphic Group Rings of Free Abelian Groups

1982 ◽  
Vol 34 (1) ◽  
pp. 8-16 ◽  
Author(s):  
Jan Krempa
Keyword(s):  
Finite Fields ◽  
Cyclic Group ◽  
Abelian Groups ◽  
Negative Answer ◽  
Partial Answer ◽  
Group Rings ◽  
Infinite Cyclic Group ◽  
Isomorphic Group ◽  
New Class ◽  
Free Abelian Groups

S. K. Sehgal ([9], Problem 26) proposed the following question : Let A, B be rings and X an infinite cyclic group. Does AX ⋍ BX imply A ⋍ B? M. M. Parmenter and S. K. Sehgal (cf. [9], Chapter 4) proved that, under some strong assumptions concerning rings A, B, the answer is affirmative. In this paper, we show that the assumptions concerning the ring B may be omitted in the above mentioned results. Moreover, it is proven that if (AX)X ⋍ BX then AX ⋍ B for all rings A, B. If A is commutative and noetherian then a partial answer to Problem 27, [9] follows from our results.Recently, L. Griinenfelder and M. M. Parmenter constructed nonisomorphic rings A, B for which the group rings AX, BX are isomorphic, [2], We give a new class of rings of this type. Our examples are noncommutative domains and algebras over finite fields. That also gives a negative answer to Problem 29, [9].

scholarly journals RIGIDITY OF GRAPH PRODUCTS OF ABELIAN GROUPS

2008 ◽  
Vol 77 (2) ◽  
pp. 187-196 ◽  
Author(s):  
MAURICIO GUTIERREZ ◽  
ADAM PIGGOTT
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Abelian Groups ◽  
Graph Products ◽  
Vertex Group ◽  
Graph Product ◽  
Finitely Generated ◽  
Product Decomposition ◽  
Unique Decomposition ◽  
Product Of Groups

AbstractWe show that if G is a group and G has a graph-product decomposition with finitely generated abelian vertex groups, then G has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely generated abelian group and the graph satisfies the T0 property. Our results build on results by Droms, Laurence and Radcliffe.

Distributivity and pseudocomplementation of lattices of generalized \(L\)-subgroups

2016 ◽  
Vol 5 (2) ◽  
pp. 107 ◽  
Author(s):  
Dilek Bayrak ◽  
Sultan Yamak
Keyword(s):  
Cyclic Group ◽  
Abelian Groups ◽  
Study Groups ◽  
Finite Cyclic Group ◽  
Pseudocomplemented Lattice ◽  
Fuzzy Subgroups

The main goal of this paper is to study the lattice of \((0,\mu)\)-\(L\)-subgroups of a group. We characterize abelian groups by the lattice of \((0,\mu)\)-\(L\)-subgroups. Also, we show that a group $G$ is locally cyclic if and only if the lattice of \((0,\mu)\)-\(L\)-subgroups is distributive. As consequence, we obtain that the lattices of all \((\in,\in\vee q)\)-fuzzy subgroups and all fuzzy subgroups of a finite cyclic group are distributive. Finally, we study groups which of the lattice of \((\lambda,\mu)\)-\(L\)-subgroups is pseudocomplemented lattice.

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.

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