Polynomials in central endomorphisms

Abstract

Let 2 be a central endomorphism of a group G in the sense that induces the identity map on the inner automorphism group of G. Despite the nearness of the situation to commutativity, it is not necessarily true that the central endomorphisms of G form a ring or even that the subset generated by be a ring. The displacement map , given by (g) -g~(g) for each g e G, is an endomorphism with central values. We shall show (Theorem 1) that if satisfies a certain pair of simultaneous equations then or z is idempotent. Let P be a formal polynomial with integral coefficients, and let t be the sum of these coefficients. Then (Theorem 2) P() is an endomorphism if and only if t induces an integral endomorphism on G. If G is nilpotent of class 2 then (Theorem 3) P() is an endomorphism if and only if t(t -l)/2 is an exponent for the commutator subgroup Q of G.