A subset $A$ if an abelian group $G$ is said to be sum-free if the sum of any pair of elements in $A$ does not belong to $A$; in other words $A$ is sum-free if there is no solution of the equation $x+y = z$ with $x,y,z in A$.

The study of sum-free subsets probably originated with a result of Schr, who showed that the set of natural numbers can not be partitioned into finitely many sum-free subsets. Using this he showed that for any positive integer $n$, the Fermat equation $x^n + y^n = z^n (mod p)$ has a nontrivial solution for all sufficiently large prime $p$. In this talk we shall see a proof, (which is rather short and not difficult) of this curious fact.

We study the question of obtaining a classification of sum-free subsets in certain special groups called type III groups and counting sum-free subsets in these groups. Our results are build upon and improve the results obtained by Ben Green and Imre Ruzsa.