Vol. 23, No. 3, 1967

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 307: 1  2
Vol. 306: 1  2
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Officers
Subscriptions
Editorial Board
Submission Guidelines
Submission Form
Contacts
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
 
Other MSP Journals
Polynomials in central endomorphisms

Franklin Haimo

Vol. 23 (1967), No. 3, 521–525
Abstract

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
Milestones
Received: 1 September 1965
Published: 1 December 1967
Authors
Franklin Haimo