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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
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 $A$.

References [Enhancements On Off] (What's this?)

  • 1. F. Anderson and K. Fuller, Rings and categories of modules, Graduate Texts in Mathematics 13 (Springer, Berlin, 1992). MR 94i:16001
  • 2. D. Carroll, ``Transitivity properties in abelian groups", doctoral thesis, Univ. Dublin, 1992.
  • 3. D. Carroll and B. Goldsmith, ``On transitive and fully transitive abelian $p$-groups", Proc. of the Royal Irish Academy (1) 96A (1996) 33-41. MR 99f:20090
  • 4. A. L. S. Corner, ``The independence of Kaplansky's notions of transitivity and full transitivity", Quart. J. Math. Oxford (2) 27 (1976) 15-20. MR 52:14090
  • 5. S. Files and B. Goldsmith, ``Transitive and fully transitive groups", Proc. Am. Math. Soc. 126 (1998) 1605-1610. MR 98g:20087
  • 6. L. Fuchs, Infinite Abelian Groups, Vol. I and II, (Academic Press, 1970 and 1973). MR 41:333, MR 50:2362
  • 7. B. Goldsmith, ``On endomorphism rings of non-separable Abelian $p$-groups", J. of Algebra 127 (1989) 73-79. MR 91b:20077
  • 8. P. Griffith, ``Transitive and fully transitive primary abelian groups", Pacific J. Math. 25 (1968) 249-254. MR 37:6374
  • 9. G. Hennecke, ``Transitivitätseigenschaften abelscher $p$-Gruppen", doctoral thesis, Essen University, 1999.
  • 10. P. Hill, ``On transitive and fully transitive primary groups", Proc. Amer. Math. Soc. 22 (1969) 414-417. MR 42:4630
  • 11. I. Kaplansky, Infinite abelian groups, (University of Michigan Press, Ann Arbor, 1954 and 1969). MR 16:444g; MR 38:2208
  • 12. B. R. Mcdonald, Finite rings with identity, (Pure and Applied Mathematics, Dekker Inc., New York, 1974). MR 50:7245
  • 13. C. Megibben, ``Large subgroups and small homomorphisms", Michigan Mathematical Journal 13 (1966) 153-160. MR 33:4135
  • 14. K. Shoda, ``Uber die Automorphismen einer endlichen Abelschen Gruppe", Math. Ann. (1928) 674-686.

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Additional Information

Agnes T. Paras
Affiliation: Department of Mathematics, University of the Philippines at Diliman, 1101 Quezon City, Philippines

Lutz Strüngmann
Affiliation: Fachbereich 6, Mathematik, University of Essen, 45117 Essen, Germany

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