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Fully transitive $p$-groups with finite first Ulm subgroup


Authors: Agnes T. Paras and Lutz Strüngmann
Journal: Proc. Amer. Math. Soc. 131 (2003), 371-377
MSC (2000): Primary 20K01, 20K10, 20K30
DOI: https://doi.org/10.1090/S0002-9939-02-06593-0
Published electronically: June 3, 2002
MathSciNet review: 1933327
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Abstract: An abelian $p$-group $G$ is called (fully) transitive if for all $x,y\in G$ with $U_G(x)=U_G(y)$ ( $U_G(x)\leq U_G(y)$) there exists an automorphism (endomorphism) of $G$ which maps $x$ onto $y$. It is a long-standing problem of A. L. S. Corner whether there exist non-transitive but fully transitive $p$-groups with finite first Ulm subgroup. In this paper we restrict ourselves to $p$-groups of type $A$, this is to say $p$-groups satisfying $\mathrm{Aut}(G)\upharpoonright_{ p^{\omega}G} = U(\mathrm{End}(G) \upharpoonright_{p^{\omega}G})$. We show that the answer to Corner's question is no if $p^{\omega}G$ is finite and $G$ is of type $A$.


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Additional Information

Agnes T. Paras
Affiliation: Department of Mathematics, University of the Philippines at Diliman, 1101 Quezon City, Philippines
Email: agnes@math01.cs.upd.edu.ph

Lutz Strüngmann
Affiliation: Fachbereich 6, Mathematik, University of Essen, 45117 Essen, Germany
Email: lutz.struengmann@uni-essen.de

DOI: https://doi.org/10.1090/S0002-9939-02-06593-0
Received by editor(s): August 9, 2001
Received by editor(s) in revised form: September 27, 2001
Published electronically: June 3, 2002
Additional Notes: The first author was supported by project No. G-0545-173,06/97 of the German-Israeli Foundation for Scientific Research & Development
The second author was supported by the Graduiertenkolleg Theoretische und Experimentelle Methoden der Reinen Mathematik of Essen University
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2002 American Mathematical Society

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