  We characterize the structure of maximum-size sum-free subsets of a random
subset of an Abelian group $G$. In particular, we determine the threshold $p_c
\approx \sqrt{\log n / n}$ above which, with high probability as $|G| \to
\infty$, each such subset is contained in a maximum-size sum-free subset of
$G$, whenever $q$ divides $|G|$ for some (fixed) prime $q$ with $q \equiv 2
\pmod 3$. Moreover, in the special case $G = \ZZ_{2n}$, we determine a sharp
threshold for the above property. The proof uses recent 'transference' theorems
of Conlon and Gowers, together with stability theorems for sum-free subsets of
Abelian groups.
