## Isolated subgroups of finite abelian groups.(English)Zbl 07547223

Summary: We say that a subgroup $$H$$ is isolated in a group $$G$$ if for every $$x\in G$$ we have either $$x\in H$$ or $$\langle x\rangle\cap H=1$$. 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.

### MSC:

 20K01 Finite abelian groups 20K27 Subgroups of abelian groups
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### References:

 [1] Amiri, H.; Jafarian Amiri, S. M.; Isaacs, I. M., Sums of element orders in finite groups, Commun. Algebra, 37, 2978-2980 (2009) · Zbl 1183.20022 [2] Busarkin, V. M., The structure of isolated subgroups in finite groups, Algebra Logika, 4, 33-50 (1965) · Zbl 0145.02904 [3] Busarkin, V. M., Groups containing isolated subgroups, Sib. Math. J., 9, 560-563 (1968) · Zbl 0192.35301 [4] Isaacs, I. M., Finite Group Theory (2008), Providence: American Mathematical Society, Providence · Zbl 1169.20001 [5] Isolated subgroup. Encyclopedia of Mathematics. Available at https://encyclopediaofmath.org/wiki/Isolated subgroup. · Zbl 1311.97001 [6] Janko, Z., Finite p-groups with some isolated subgroups, J. Algebra, 465, 41-61 (2016) · Zbl 1354.20012 [7] Kurosh, A. G., The Theory of Groups (1960), New York: Chelsea Publishing, New York · Zbl 0094.24501 [8] Suzuki, M., Group Theory. I (1982), Berlin: Springer, Berlin [9] Tărnăuceanu, M., A generalization of a result on the sum of element orders of a finite group, Math. Slovaca, 71, 627-630 (2021) · Zbl 07438366 [10] Tărnăuceanu, M.; Fodor, D. G., On the sum of element orders of finite Abelian groups, An. Ştiimţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat., 60, 1-7 (2014) · Zbl 1299.20059
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