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Finally $S$ fills $G$ if $G^*\\subseteq S \\sqcup SS$ (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 (in `Group Ramsey Theory', J. Comb. Theory Series A, 17 (1974) 219-226) investigated filled groups and gave a classification of filled abelian groups. 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