More on groups in which each element commutes with its endomorphic images

Abstract

R. Faudree has given examples of nonabelian groups which have the property cited in the title. His groups are p-groups such that (i) Z ( G ) = G ′ = G p = U ( G ) Z(G) = G’ = {G^p} = U(G) , (ii) each endomorphism ϕ \phi (which is not an automorphism) has ( G ) ϕ ⩽ Z ( G ) (G)\phi \leqslant Z(G) , and (iii) each automorphism is central. In this paper the necessity of these conditions is explored. It is also shown that, for p = 2, Faudree’s example does not in fact have the property cited in the title.