Centers of centralizer nearrings determined by All endomorphisms of symmetric groups

For n = ۵, ۶ and E = \End S_n, the functions in the centralizer nearring M_E(S_n) = \{f : S_n \to S_n \ |\ f(۱) = (۱) \ \hbox{and} \ f \circ s = s \circ f \ \hbox{for all}\ s \in E\} are determined.  The centers of these two nearrings are also described.  Results that can be used to determine the functions in M_E(S_n) and their centers for n \geq ۷ are also presented.

Keywords:

Centralizer nearring , symmetric group , center , outer automorphism