scholarly journals Maximal sum-free sets in abelian groups of order divisible by three

1972 ◽  
Vol 6 (3) ◽  
pp. 439-441 ◽  
Author(s):  
Anne Penfold Street
Keyword(s):  
Abelian Groups ◽  
Additive Group ◽  
Finite Abelian Groups ◽  
Free Set ◽  
Free Sets

A subset S of an additive group G is called a maximal sum-free set in G if (S+S) nS = Φ and |S| ≥ |T| for every sum-free set T in G. In this note, we prove a conjecture of Yap concerning the structure of maximal sum-free sets in finite abelian groups of order divisible by 3 but not divisible by any prime congruent to 2 modulo 3.

Maximal sum-free sets in finite abelian groups

1970 ◽  
Vol 2 (3) ◽  
pp. 289-297 ◽  
Author(s):  
A. H. Rhemtulla ◽  
Anne Penfold Street
Keyword(s):  
Abelian Groups ◽  
Additive Group ◽  
Finite Abelian Groups ◽  
Free Set ◽  
Free Sets

A subset S of an additive group G is called a maximal sum-free set in G if (S+S) ∩ S = ø and ∣S∣ ≥ ∣T∣ for every sum-free set T in G. It is shown that if G is an elementary abelian p–group of order pn, where p = 3k ± 1, then a maximal sum-free set in G has kpn-1 elements. The maximal sum-free sets in Zp are characterized to within automorphism.

scholarly journals Maximal sum-free sets in finite abelian groups, V

1975 ◽  
Vol 13 (3) ◽  
pp. 337-342 ◽  
Author(s):  
H.P. Yap
Keyword(s):  
Prime Divisor ◽  
Abelian Groups ◽  
Finite Abelian Groups ◽  
Free Set ◽  
Free Sets

Let λ(G) be the cardinality of a maximal sum-free set in a group G. Diananda and Yap conjectured that if G is abelian and if every prime divisor of |G| is congruent to 1 modulo 3, then λ(G) = |G|(n−1)/3n where n is the exponent of G. This conjecture has been proved to be true for elementary abelian p−groups by Rhemtulla and Street ana for groups by Yap. We now prove this conjecture for groups G = Zpq ⊕ Zp where p and q are distinct primes.

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.

scholarly journals Zero square near-rings

1991 ◽  
Vol 51 (3) ◽  
pp. 497-504
Author(s):  
Patricia Jones
Keyword(s):  
Abelian Groups ◽  
Additive Group ◽  
Finite Abelian Groups

AbstractThe purpose of this paper is to provide examples and explore properties of a wide variety of zero square (left) near rings. Among the main results are complete classifications of (i) finite Abelian groups which are the additive group of a zero square near-ring and (ii) finite non-Abelian groups which support 3-nilpotent distributive zero square near-rings.

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 Maximal sum-free sets in abelian groups of order divisible by three: Corrigendum

1972 ◽  
Vol 7 (2) ◽  
pp. 317-318 ◽  
Author(s):  
Anne Penfold Street
Keyword(s):  
Arithmetic Progression ◽  
Abelian Groups ◽  
Factor Group ◽  
Image Position ◽  
Free Set ◽  
Free Sets

The last step of the proof in [2] was omitted. To complete the argument, we proceed in the following way. We had shown that H = H(S) = H(S+S) = H(S-S), that |S-S| = 2|S| - |H| and hence that in the factor group G* = G/H of order 3m, the maximal sum-free set S* = S/H and its set of differences S* - S* are aperiodic, withso thatBy (1) and Theorem 2.1 of [1], S* - S* is either quasiperiodic or in arithmetic progression.

scholarly journals Maximal sum-free sets in finite abelian groups, II

1971 ◽  
Vol 5 (1) ◽  
pp. 43-54 ◽  
Author(s):  
H.P. Yap
Keyword(s):  
Positive Integer ◽  
Prime Divisor ◽  
Abelian Groups ◽  
Finite Abelian Groups ◽  
Free Sets

Maximal sum-free sets in groups Zn, where n is any positive integer such, that every prime divisor of n is congruent to 1 modulo 3, are completely characterized.

Maximal Sum-Free Sets in Elementary Abelian p-Groups

1971 ◽  
Vol 14 (1) ◽  
pp. 73-80 ◽  
Author(s):  
A. H. Rhemtulla ◽  
Anne Penfold Street
Keyword(s):  
Additive Group ◽  
Free Set ◽  
Free Sets

Given an additive group G and nonempty subsets S, T of G, let S+T denote the set ﹛s + t | s ∊ S, t ∊ T﹜, S the complement of S in G and |S| the cardinality of S. We call S a sum-free set in G if (S+S) ⊆ S. If, in addition, |S| ≥ |T| for every sum-free set T in G, then we call S a maximal sum-free set in G. We denote by λ(G) the cardinality of a maximal sum-free set in G.

scholarly journals Maximal sum-free sets in cyclic groups of prime-power order

1971 ◽  
Vol 4 (3) ◽  
pp. 407-418
Author(s):  
Anne Penfold Street
Keyword(s):  
Prime Power ◽  
Additive Group ◽  
Prime Power Order ◽  
Cyclic Groups ◽  
Free Set ◽  
Free Sets

A subset S of an additive group G is called a maximal sum-free set in G if (S+S) ∩ S = ø and |S| ≥ |T| for every sum-free set T in G. In this paper, the maximal sum-free sets in cyclic p–groups are characterized to within automorphism.

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