2015/06/08 by Chimere S. Anabanti, Anabanti, Chimere S., Sarah B. Hart +1
Computer Science · Mathematics · #Advanced Graph Theory Research #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Limits and Structures in Graph Theory #Primary 20D60 #Secondary 20P05
paper · pdf · doi:10.48550/arxiv.1506.02430
openalex publication_date 2015/06/08 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
Let S be a non-empty subset of a group G. We say S is product-free if\nS\∩ SS= varnothing, and S is locally maximal if whenever T is\nproduct-free and S\⊆ T, then S=T. Finally S fills G if\nG^*\⊆ S sqcup SS (where G^* is the set of all non-identity elements\nof G), and G is a filled group if every locally maximal product-free set in\nG fills G. Street and Whitehead (in `Group Ramsey Theory', J. Comb. Theory\nSeries A, 17 (1974) 219-226) investigated filled groups and gave a\nclassification of filled abelian groups. In this paper, we obtain some results\nabout filled groups in the non-abelian case, including a classification of\nfilled groups of odd order. Street and Whitehead conjectured that the finite\ndihedral group of order 2n is not filled when n=6k+1 (k\≥ 1). We\ndisprove this conjecture on dihedral groups, and in doing so obtain a\nclassification of locally maximal product-free sets of sizes 3 and 4 in\ndihedral groups.\n