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Projectively Invariant Subgroups of Abelian \(p\)-Groups

Abstract

For a projectively invariant subgroup \(C\) of a reduced \(p\)-group \(G\), a nondecreasing sequence of ordinals and the symbol \(\infty\) is constructed in which the \(k\)th position, \(k=0,1,2,\dots\), is occupied by the minimum of heights in \(G\) of all nonzero elements of the subgroup \(p^kC[p]\). It is proved that if all elements of this sequence are integers, then the subgroup \(C\) is fully invariant.

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References

  1. 1

    C. Megibben, “Projection-invariant subgroups of Abelian groups,” Tamkang J. Math. 8 (2), 177–182 (1977).

    MathSciNet  MATH  Google Scholar 

  2. 2

    J. Hausen, “Endomorphism rings generated by idempotents,” Tamkang J. Math. 12 (2), 215–218 (1981).

    MathSciNet  MATH  Google Scholar 

  3. 3

    A. R. Chekhlov, “On projective invariant subgroups of Abelian groups,” J. Math. Sci. 164 (1), 143–147 (2010).

    MathSciNet  Article  Google Scholar 

  4. 4

    P. Danchev and B. Goldsmith, “On projective invariant subgroups of Abelian \(p\)-groups,” in Groups and Model Theory, Contemp. Math. (Amer. Math. Soc., Providence, RI, 2012), Vol. 576, pp. 31–40.

    Article  Google Scholar 

  5. 5

    L. Fuchs, Infinite Abelian Groups (Academic Press, New York–London, 1970), Vol. 1.

    MATH  Google Scholar 

  6. 6

    A. L. S. Corner, “On endomorphism rings of primary abelian groups. II,” Quart. J. Math. Oxford Ser. (2) 27 (2), 5–13 (1976).

    MathSciNet  Article  Google Scholar 

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Correspondence to A. R. Chekhlov.

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Chekhlov, A.R. Projectively Invariant Subgroups of Abelian \(p\)-Groups. Math Notes 109, 948–953 (2021). https://doi.org/10.1134/S000143462105028X

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Keywords

  • projection
  • projectively invariant subgroup
  • fully invariant subgroup