On semi-endomorphisms of groups

Given a group G, a mapping α : G → G is said to be a semi-endomorphism of G if α(x + y + x) = α(x) + α(y) + α(x) for all x,y ∈ G. It is shown that any non-trivial 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 ≥ 4, are described. As an application, it is proved that the semi-endomorphism nearring S(S_n) of S_n, with n ≥ 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.