Vol. 23, No. 3, 1967

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Polynomials in central endomorphisms

Franklin Haimo

Vol. 23 (1967), No. 3, 521–525

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) = g1λ(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.

Mathematical Subject Classification
Primary: 20.22
Received: 1 September 1965
Published: 1 December 1967
Franklin Haimo