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...

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Main Authors:	Lingling Han, Xiuyun Guo
Format:	Article
Language:	English
Published:	MDPI AG 2020-09-01
Series:	Symmetry
Subjects:	subgroup chain fuzzy subgroup cyclic group elementary abelian group
Online Access:	https://www.mdpi.com/2073-8994/12/9/1537

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spelling	doaj-3c14e3a402f7423d9d185eea7c9598b92020-11-25T03:07:16ZengMDPI AGSymmetry2073-89942020-09-01121537153710.3390/sym12091537The Number of Subgroup Chains of Finite Nilpotent GroupsLingling Han0Xiuyun Guo1Department of Mathematics, Shanghai University, Shanghai 200444, ChinaDepartment of Mathematics, Shanghai University, Shanghai 200444, ChinaIn 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.https://www.mdpi.com/2073-8994/12/9/1537subgroup chainfuzzy subgroupcyclic groupelementary abelian group
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Lingling Han Xiuyun Guo
spellingShingle	Lingling Han Xiuyun Guo The Number of Subgroup Chains of Finite Nilpotent Groups Symmetry subgroup chain fuzzy subgroup cyclic group elementary abelian group
author_facet	Lingling Han Xiuyun Guo
author_sort	Lingling Han
title	The Number of Subgroup Chains of Finite Nilpotent Groups
title_short	The Number of Subgroup Chains of Finite Nilpotent Groups
title_full	The Number of Subgroup Chains of Finite Nilpotent Groups
title_fullStr	The Number of Subgroup Chains of Finite Nilpotent Groups
title_full_unstemmed	The Number of Subgroup Chains of Finite Nilpotent Groups
title_sort	number of subgroup chains of finite nilpotent groups
publisher	MDPI AG
series	Symmetry
issn	2073-8994
publishDate	2020-09-01
description	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.
topic	subgroup chain fuzzy subgroup cyclic group elementary abelian group
url	https://www.mdpi.com/2073-8994/12/9/1537
work_keys_str_mv	AT linglinghan thenumberofsubgroupchainsoffinitenilpotentgroups AT xiuyunguo thenumberofsubgroupchainsoffinitenilpotentgroups AT linglinghan numberofsubgroupchainsoffinitenilpotentgroups AT xiuyunguo numberofsubgroupchainsoffinitenilpotentgroups
_version_	1724671519310741504

The Number of Subgroup Chains of Finite Nilpotent Groups

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