Locally maximal product - free sets of size 3
نویسندگان
چکیده
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 were classified in [3]. In this paper, we prove a conjecture of Giudici and Hart in [3] by showing that if S is a locally maximal product-free set of size 3 in a group G, then |G| ≤ 24. This shows that the list of known locally maximal product-free sets given in [3] is complete.
منابع مشابه
On a conjecture of Street and Whitehead on locally maximal product-free sets
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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