The Subgroups of Finite Metacyclic Groups

Xu Yang

Chinese Annals of Mathematics, Series B ›› 2020, Vol. 41 ›› Issue (2) : 241 -266.

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Chinese Annals of Mathematics, Series B ›› 2020, Vol. 41 ›› Issue (2) : 241 -266. DOI: 10.1007/s11401-020-0197-6
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The Subgroups of Finite Metacyclic Groups

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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.

Keywords

Metacyclic groups / Subgroups / Metacyclic p-groups / Characteristic subgroups

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Xu Yang. The Subgroups of Finite Metacyclic Groups. Chinese Annals of Mathematics, Series B, 2020, 41(2): 241-266 DOI:10.1007/s11401-020-0197-6

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