  Let $S$ be a non-empty subset of a group $G$. We say $S$ is product-free if
$S\cap SS=\varnothing$, and $S$ is locally maximal if whenever $T$ is
product-free and $S\subseteq T$, then $S=T$. 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. In this paper, we obtain some results
about filled groups in the non-abelian case, including a classification of
filled groups of odd order. Street and Whitehead conjectured that the finite
dihedral group of order $2n$ is not filled when $n=6k+1$ ($k\geq 1$). We
disprove this conjecture on dihedral groups, and in doing so obtain a
classification of locally maximal product-free sets of sizes 3 and 4 in
dihedral groups.
