Orthomorphism

In abstract algebra, an orthomorphism is a certain kind of mapping from a group into itself. Let "G" be a group, and let "θ" be a permutation of "G". Then "θ" is an orthomorphism of "G" if the mapping "f" defined by "f"("x") = "x"−1 "θ"("x") is also a permutation of "G". A permutation "φ" of "G" is a complete mapping if the mapping "g" defined by "g"("x") = "xφ"("x") is also a permutation of "G". Orthomorphisms and complete mappings are closely related. 