Detecting the index of a subgroup in the subgroup lattice
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- by M. De Falco, F. de Giovanni, C. Musella and R. Schmidt
- Proc. Amer. Math. Soc. 133 (2005), 979-985
- DOI: https://doi.org/10.1090/S0002-9939-04-07638-5
- Published electronically: September 16, 2004
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Abstract:
A theorem by Zacher and Rips states that the finiteness of the index of a subgroup can be described in terms of purely lattice-theoretic concepts. On the other hand, it is clear that if $G$ is a group and $H$ is a subgroup of finite index of $G$, the index $|G:H|$ cannot be recognized in the lattice ${\mathfrak {L}}(G)$ of all subgroups of $G$, as for instance all groups of prime order have isomorphic subgroup lattices. The aim of this paper is to give a lattice-theoretic characterization of the number of prime factors (with multiplicity) of $|G:H|$.References
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Bibliographic Information
- M. De Falco
- Affiliation: Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, Via Cintia, I - 80126 Napoli, Italy
- Email: mdefalco@unina.it
- F. de Giovanni
- Affiliation: Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, Via Cintia, I - 80126 Napoli, Italy
- Email: degiovan@unina.it
- C. Musella
- Affiliation: Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, Via Cintia, I - 80126 Napoli, Italy
- Email: cmusella@unina.it
- R. Schmidt
- Affiliation: Mathematisches Seminar, Universität Kiel, Ludwig-Meyn Straße 4, D - 24098 Kiel, Germany
- Email: schmidt@math.uni-kiel.de
- Received by editor(s): October 8, 2003
- Received by editor(s) in revised form: December 1, 2003
- Published electronically: September 16, 2004
- Communicated by: Jonathan I. Hall
- © Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 979-985
- MSC (2000): Primary 20E15
- DOI: https://doi.org/10.1090/S0002-9939-04-07638-5
- MathSciNet review: 2117197