Abstract
In this paper, the author characterizes the subgroups of a finite metacyclic group K by building a one to one correspondence between certain 3-tuples (k,l,β) ∈ ℕ3 and all the subgroups of K. The results are applied to compute some subgroups of K as well as to study the structure and the number of p-subgroups of K, where p is a fixed prime number. In addition, the author gets a factorization of K, and then studies the metacyclic p-groups, gives a different classification, and describes the characteristic subgroups of a given metacyclic p-group when p ≥ 3. A “reciprocity” relation on enumeration of subgroups of a metacyclic group is also given.
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Acknowledgement
The author is grateful to Professor Zhu Shenglin for his guidance during the work.
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This work was supported by the National Natural Science Foundation of China (No. 11331006).
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Yang, X. The Subgroups of Finite Metacyclic Groups. Chin. Ann. Math. Ser. B 41, 241–266 (2020). https://doi.org/10.1007/s11401-020-0197-6
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DOI: https://doi.org/10.1007/s11401-020-0197-6