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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Fully transitive $p$-groups with finite first Ulm subgroup
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by Agnes T. Paras and Lutz StrĂĽngmann PDF
Proc. Amer. Math. Soc. 131 (2003), 371-377 Request permission

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 StrĂĽngmann
  • Affiliation: Fachbereich 6, Mathematik, University of Essen, 45117 Essen, Germany
  • Email: lutz.struengmann@uni-essen.de
  • 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
  • © Copyright 2002 American Mathematical Society
  • 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
  • MathSciNet review: 1933327