  A finite abelian group $G$ of cardinality $n$ is said to be of type III if
every prime divisor of $n$ is congruent to 1 modulo 3. We obtain a
classification theorem for sum-free subsets of largest possible cardinality in
a finite abelian group $G$ of type III. This theorem, when taken together with
known results, gives a complete characterisation of sum-free subsets of the
largest cardinality in any finite abelian group $G$. We supplement this result
with a theorem on the structure of sum-free subsets of cardinality "close" to
the largest possible in a type III abelian group $G$. We then give two
applications of these results. Our first application allows us to write down a
formula for the number of orbits under the natural action of ${\rm Aut}(G)$ on
the set of sum-free subsets of $G$ of the largest cardinality when $G$ is of
the form $({\mathbf{Z}}/m{\mathbf{Z}})^r$, with all prime divisors of $m$
congruent to 1 modulo 3, thereby extending a result of Rhemtulla and Street.
Our second application provides an upper bound for the number of sum-free
subsets of $G$. For finite abelian groups $G$ of type III and with {\em a given
exponent} this bound is substantially better than that implied by the bound for
the number of sum-free subsets in an arbitrary finite abelian group, due to
Green and Ruzsa.
