The Number of Subgroup Chains of Finite Nilpotent Groups
In this paper, we mainly count the number of subgroup chains of a finite nilpotent group. We derive a recursive formula that reduces the counting problem to that of finite <i>p</i>-groups. As applications of our main result, the classification problem of distinct fuzzy subgroups of finit...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-09-01
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| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/12/9/1537 |
| Summary: | In this paper, we mainly count the number of subgroup chains of a finite nilpotent group. We derive a recursive formula that reduces the counting problem to that of finite <i>p</i>-groups. As applications of our main result, the classification problem of distinct fuzzy subgroups of finite abelian groups is reduced to that of finite abelian <i>p</i>-groups. In particular, an explicit recursive formula for the number of distinct fuzzy subgroups of a finite abelian group whose Sylow subgroups are cyclic groups or elementary abelian groups is given. |
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| ISSN: | 2073-8994 |