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| The Subgroups of Finite Metacyclic Groups |
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Citation： |
Xu YANG.The Subgroups of Finite Metacyclic Groups[J].Chinese Annals of Mathematics B,2020,41(2):241~266 |
| Page view： 793
Net amount： 671 |
Authors： |
Xu YANG; |
Foundation： |
This work was supported by the National Natural Science
Foundation of China (No.11331006). |
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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 $(k,l,\beta)\in \mathbb{N}^3$ 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 $p\geq3$. A
``reciprocity'' relation on enumeration of subgroups of a metacyclic
group is also given. |
Keywords： |
Metacyclic groups, Subgroups, Metacyclic $p$-groups, Characteristic subgroups |
Classification： |
20D15, 20D25, 20D30 |
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