Let λ be a central endomorphism
of a group G in the sense that λ 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 λ be a ring. The displacement map τ, given by
τ(g) = g−1λ(g) for each g ∈ G, is an endomorphism with central values. We shall
show (Theorem 1) that if τ satisfies a certain pair of simultaneous equations
then λ or λ2 is idempotent. Let P be a formal polynomial with integral
coefficients, and let t be the sum of these coefficients. Then (Theorem 2) P(λ) 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(λ) is an endomorphism if
and only if t(t − 1)∕2 is an exponent for the commutator subgroup Q of
G.
