On semi-endomorphisms of groups

Given a group G, a mapping alpha: G --> G is said to be a semi-endomorphism of G if alpha(x + y + x) = alpha(x) + alpha(y) + a(x) for all x,y is an element of G. It is shown that any nontrivial zero preserving semi-endomorphism of a finite simple group of order greater : than two is either an automorphism or an anti-automorphism. Moreover, the semi-endomorphisms of S-n, the symmetric group of degree n, n greater than or equal to 4, are described. As an application, it is proved that the semi-endomorphism nearring S(S-n) of S-n with n greater than or equal to 3 is equal to E(S-n) + M-c(S-n), where E(S-n) is the endomorphism nearring of S-n, and M-c(S-n) is the nearring of constant mappings of S-n.