Abstract
We say that a subgroup H is isolated in a group G if for every x ∈ G we have either x ∈ H or 〈x〉 ∩ H = 1. We describe the set of isolated subgroups of a finite abelian group. The technique used is based on an interesting connection between isolated subgroups and the function sum of element orders of a finite group.
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References
H. Amiri, S. M. Jafarian Amiri, I. M. Isaacs: Sums of element orders in finite groups. Commun. Algebra 37 (2009), 2978–2980.
V. M. Busarkin: The structure of isolated subgroups in finite groups. Algebra Logika 4 (1965), 33–50. (In Russian.)
V. M. Busarkin: Groups containing isolated subgroups. Sib. Math. J. 9 (1968), 560–563.
I. M. Isaacs: Finite Group Theory. Graduate Studies in Mathematics 92. American Mathematical Society, Providence, 2008.
Isolated subgroup. Encyclopedia of Mathematics. Available at https://encyclopediaofmath.org/wiki/Isolated subgroup.
Z. Janko: Finite p-groups with some isolated subgroups. J. Algebra 465 (2016), 41–61.
A. G. Kurosh: The Theory of Groups. Volume I, II. Chelsea Publishing, New York, 1960.
M. Suzuki: Group Theory. I. Grundlehren der Mathematischen Wissenschaften 247. Springer, Berlin, 1982.
M. Tărnăuceanu: A generalization of a result on the sum of element orders of a finite group. Math. Slovaca 71 (2021), 627–630.
M. Tărnăuceanu, D. G. Fodor: On the sum of element orders of finite Abelian groups. An. Ştiimţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 60 (2014), 1–7.
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The author is grateful to the reviewer for remarks which improved the previous version of the paper.
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Tărnăuceanu, M. Isolated subgroups of finite abelian groups. Czech Math J 72, 615–620 (2022). https://doi.org/10.21136/CMJ.2022.0085-21
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DOI: https://doi.org/10.21136/CMJ.2022.0085-21