Morphic groups

A group G is called morphic if every endomorphism α : G → G for which G α is normal in G satisfies G / G α ≅ k e r ( α ) . This concept originated in a 1976 paper of Gertrude Ehrlich characterizing when the endomorphism ring of a module is unit regular. The concept has been extensively studied in mo...

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Bibliographic Details
Published inJournal of pure and applied algebra Vol. 214; no. 10; pp. 1827 - 1834
Main Authors Li, Yuanlin, Nicholson, W.K., Zan, Libo
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.10.2010
Subjects
20D45
20F06
20F28
16A30
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Summary:A group G is called morphic if every endomorphism α : G → G for which G α is normal in G satisfies G / G α ≅ k e r ( α ) . This concept originated in a 1976 paper of Gertrude Ehrlich characterizing when the endomorphism ring of a module is unit regular. The concept has been extensively studied in module and ring theory, and this paper investigates the idea in the category of groups. After developing their basic properties, we characterize the morphic groups among the dihedral groups and the groups whose normal subgroups form a finite chain. We investigate when a direct product of morphic groups is again morphic, prove that a finite nilpotent group is morphic if and only if its Sylow subgroups are morphic, and present some results for the case where a p -group is morphic.
ISSN:0022-4049
1873-1376
DOI:10.1016/j.jpaa.2009.12.026