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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On automorphism groups and endomorphism rings of abelian $ p$-groups


Author: Jutta Hausen
Journal: Trans. Amer. Math. Soc. 210 (1975), 123-128
MSC: Primary 20K30
MathSciNet review: 0376906
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Abstract: Let $ A$ be a noncyclic abelian $ p$-group where $ p \geqslant 5$, and let $ {p^\infty }A$ be the maximal divisible subgroup of $ A$. It is shown that $ A/{p^\infty }A$ is bounded and nonzero if and only if the automorphism group of $ A$ contains a minimal noncentral normal subgroup. This leads to the following connection between the ideal structure of certain rings and the normal structure of their groups of units: if the noncommutative ring $ R$ is isomorphic to the full ring of endomorphisms of an abelian $ p$-group, $ p \geqslant 5$, then $ R$ contains minimal twosided ideals if and only if the group of units of $ R$ contains minimal noncentral normal subgroups.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0376906-9
Keywords: Abelian $ p$-group, automorphism group, normal subgroups of automorphism groups, endomorphism ring
Article copyright: © Copyright 1975 American Mathematical Society