Let S be a non-empty subset of a group G. We say S is product-free if
the equation xy=z does not hold for x,y,z\in S; equivalently, if SnSS=Ø.
A product-free set S is locally maximal if whenever T is product-free and
S\subseteq T, then S=T. Finally, a product-free set S fills G if
G*\subseteq S \cup SS? (where G*=G\{1}?), and G is called a filled group
if every locally maximal product-free set in G fills G. The study of filled groups
started in 1974 with Street and Whitehead. They classified all finite abelian
filled groups, and conjectured that the finite dihedral group of order 2n
is not filled when n=6k+1 (k>0). In this talk, we disprove this conjecture.
Moreover, we obtain a classification of filled groups of different sorts, including
filled dihedral groups, filled groups of odd order, and filled nilpotent groups.
(Joint work with Sarah Hart and Grahame Erskine)