Groups Containing Small Locally Maximal Product-Free Sets
نویسندگان
چکیده
منابع مشابه
Groups whose locally maximal product - free sets are complete
Let G be a finite group and S a subset of G. Then S is product-free if S ∩ SS = ∅, and complete if G∗ ⊆ S ∪ SS. A product-free set is locally maximal if it is not contained in a strictly larger product-free set. If S is product-free and complete then S is locally maximal, but the converse does not necessarily hold. Street and Whitehead [11] defined a group G as filled if every locally maximal p...
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Let G be a group, and S a non-empty subset of G. Then S is product-free if ab / ∈ S for all a, b ∈ S. We say S is locally maximal product-free if S is product-free and not properly contained in any other product-free set. A natural question is what is the smallest possible size of a locally maximal product-free set in G. The groups containing locally maximal product-free sets of sizes 1 and 2 w...
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Let G be a group and S a non-empty subset of G. If ab / ∈ S for any a, b ∈ S, then S is called sum-free. We show that if S is maximal by inclusion and no proper subset generates 〈S〉 then |S| ≤ 2. We determine all groups with a maximal (by inclusion) sum-free set of size at most 2 and all of size 3 where there exists a ∈ S such that a / ∈ 〈S \ {a}〉.
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Let S be a non-empty subset of a group G. We say S is product-free if S ∩ SS = ∅, and S is locally maximal if whenever T is product-free and S ⊆ T , then S = T . Finally S fills G if G∗ ⊆ S t SS (where G∗ is the set of all non-identity elements of G), and G is a filled group if every locally maximal product-free set in G fills G. Street and Whitehead [8] investigated filled groups and gave a cl...
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This paper studies the maximal size of product-free sets in Z/nZ. These are sets of residues for which there is no solution to ab ≡ c (mod n) with a, b, c in the set. In a previous paper we constructed an infinite sequence of integers (ni)i≥1 and product-free sets Si in Z/niZ such that the density |Si|/ni → 1 as i → ∞, where |Si| denotes the cardinality of Si. Here we obtain matching, up to con...
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ژورنال
عنوان ژورنال: International Journal of Combinatorics
سال: 2016
ISSN: 1687-9163,1687-9171
DOI: 10.1155/2016/8939182